# Half-Open Rectangle and Unit-Cell Area Foundations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261005-advance-20261006-r06-area-foundations · 2026-10-05 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,0)×[0,3) has width 0 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation0

Multiply width by height to obtain 0 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0001/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 2. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,1)×[0,2) has width 1 and height 2, both nonnegative. Find its area in squared length units.

View answer and explanation2

Multiply width by height to obtain 2 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0002/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 3. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,2)×[0,3) has width 2 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation6

Multiply width by height to obtain 6 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0003/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 4. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,3)×[0,4) has width 3 and height 4, both nonnegative. Find its area in squared length units.

View answer and explanation12

Multiply width by height to obtain 12 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0004/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 5. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,4)×[0,5) has width 4 and height 5, both nonnegative. Find its area in squared length units.

View answer and explanation20

Multiply width by height to obtain 20 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0005/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 6. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,5)×[0,1) has width 5 and height 1, both nonnegative. Find its area in squared length units.

View answer and explanation5

Multiply width by height to obtain 5 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0006/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 7. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,6)×[0,2) has width 6 and height 2, both nonnegative. Find its area in squared length units.

View answer and explanation12

Multiply width by height to obtain 12 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0007/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 8. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,7)×[0,3) has width 7 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation21

Multiply width by height to obtain 21 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0008/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 9. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,8)×[0,4) has width 8 and height 4, both nonnegative. Find its area in squared length units.

View answer and explanation32

Multiply width by height to obtain 32 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0009/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 10. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,9)×[0,5) has width 9 and height 5, both nonnegative. Find its area in squared length units.

View answer and explanation45

Multiply width by height to obtain 45 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0010/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 11. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,10)×[0,1) has width 10 and height 1, both nonnegative. Find its area in squared length units.

View answer and explanation10

Multiply width by height to obtain 10 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0011/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 12. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,11)×[0,2) has width 11 and height 2, both nonnegative. Find its area in squared length units.

View answer and explanation22

Multiply width by height to obtain 22 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0012/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 13. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,12)×[0,3) has width 12 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation36

Multiply width by height to obtain 36 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0013/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 14. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,13)×[0,4) has width 13 and height 4, both nonnegative. Find its area in squared length units.

View answer and explanation52

Multiply width by height to obtain 52 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0014/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 15. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,14)×[0,5) has width 14 and height 5, both nonnegative. Find its area in squared length units.

View answer and explanation70

Multiply width by height to obtain 70 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0015/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 16. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,15)×[0,1) has width 15 and height 1, both nonnegative. Find its area in squared length units.

View answer and explanation15

Multiply width by height to obtain 15 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0016/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 17. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,16)×[0,2) has width 16 and height 2, both nonnegative. Find its area in squared length units.

View answer and explanation32

Multiply width by height to obtain 32 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0017/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 18. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,17)×[0,3) has width 17 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation51

Multiply width by height to obtain 51 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0018/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 19. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,18)×[0,4) has width 18 and height 4, both nonnegative. Find its area in squared length units.

View answer and explanation72

Multiply width by height to obtain 72 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0019/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 20. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,19)×[0,5) has width 19 and height 5, both nonnegative. Find its area in squared length units.

View answer and explanation95

Multiply width by height to obtain 95 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0020/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 21. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,20)×[0,1) has width 20 and height 1, both nonnegative. Find its area in squared length units.

View answer and explanation20

Multiply width by height to obtain 20 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0021/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 22. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,21)×[0,2) has width 21 and height 2, both nonnegative. Find its area in squared length units.

View answer and explanation42

Multiply width by height to obtain 42 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0022/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 23. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,22)×[0,3) has width 22 and height 3, both nonnegative. Find its area in squared length units.

View answer and explanation66

Multiply width by height to obtain 66 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0023/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 24. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,23)×[0,4) has width 23 and height 4, both nonnegative. Find its area in squared length units.

View answer and explanation92

Multiply width by height to obtain 92 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0024/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 25. Area of a Half-Open Rectangle with Nonnegative Dimensions

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,24)×[0,5) has width 24 and height 5, both nonnegative. Find its area in squared length units.

View answer and explanation120

Multiply width by height to obtain 120 squared units. A zero extent makes the half-open rectangle empty; this is area, not perimeter.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0025/)

Exact area of half-open rectangles and unit-cell unions · qf-area-rectangle-dimensions · candidate





## 26. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [0, 0)×[-2, 3), find the exact area in squared length units.

View answer and explanation0

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 0 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0026/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 27. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-5, -3)×[-11, -9), find the exact area in squared length units.

View answer and explanation4

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 4 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0027/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 28. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-6, -3)×[-10, -7), find the exact area in squared length units.

View answer and explanation9

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 9 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0028/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 29. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-7, -3)×[-9, -5), find the exact area in squared length units.

View answer and explanation16

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 16 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0029/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 30. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-8, -7)×[-8, -3), find the exact area in squared length units.

View answer and explanation5

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 5 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0030/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 31. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-9, -7)×[-7, -1), find the exact area in squared length units.

View answer and explanation12

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 12 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0031/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 32. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-10, -7)×[-6, 1), find the exact area in squared length units.

View answer and explanation21

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 21 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0032/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 33. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-11, -7)×[-5, 3), find the exact area in squared length units.

View answer and explanation32

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 32 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0033/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 34. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-12, -11)×[-4, 5), find the exact area in squared length units.

View answer and explanation9

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 9 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0034/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 35. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-13, -11)×[-3, 7), find the exact area in squared length units.

View answer and explanation20

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 20 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0035/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 36. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-14, -11)×[-2, 9), find the exact area in squared length units.

View answer and explanation33

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 33 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0036/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 37. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-15, -11)×[-1, 11), find the exact area in squared length units.

View answer and explanation48

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 48 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0037/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 38. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-16, -15)×[0, 13), find the exact area in squared length units.

View answer and explanation13

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 13 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0038/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 39. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-17, -15)×[1, 15), find the exact area in squared length units.

View answer and explanation28

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 28 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0039/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 40. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-18, -15)×[2, 17), find the exact area in squared length units.

View answer and explanation45

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 45 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0040/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 41. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-19, -15)×[3, 19), find the exact area in squared length units.

View answer and explanation64

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 64 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0041/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 42. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-20, -19)×[4, 21), find the exact area in squared length units.

View answer and explanation17

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 17 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0042/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 43. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-21, -19)×[5, 23), find the exact area in squared length units.

View answer and explanation36

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 36 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0043/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 44. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-22, -19)×[6, 25), find the exact area in squared length units.

View answer and explanation57

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 57 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0044/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 45. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-23, -19)×[7, 27), find the exact area in squared length units.

View answer and explanation80

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 80 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0045/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 46. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-24, -23)×[8, 29), find the exact area in squared length units.

View answer and explanation21

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 21 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0046/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 47. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-25, -23)×[9, 31), find the exact area in squared length units.

View answer and explanation44

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 44 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0047/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 48. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-26, -23)×[10, 33), find the exact area in squared length units.

View answer and explanation69

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 69 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0048/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 49. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-27, -23)×[11, 35), find the exact area in squared length units.

View answer and explanation96

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 96 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0049/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 50. Area of a Half-Open Integer-Endpoint Rectangle

Use one equal-unit orthonormal Euclidean Cartesian frame. For the half-open rectangle [-28, -27)×[12, 37), find the exact area in squared length units.

View answer and explanation25

Subtract each lower endpoint from its upper endpoint and multiply the two nonnegative extents, obtaining 25 squared units. A zero width or height makes the rectangle empty.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0050/)

Exact area of half-open rectangles and unit-cell unions · qf-area-coordinate-rectangle · candidate





## 51. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as []; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation0

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 0 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0051/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 52. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation1

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 1 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0052/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 53. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation2

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 2 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0053/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 54. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation3

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 3 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0054/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 55. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation4

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 4 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0055/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 56. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation5

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 5 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0056/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 57. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation6

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 6 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0057/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 58. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation7

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 7 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0058/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 59. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation8

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 8 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0059/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 60. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation9

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 9 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0060/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 61. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation10

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 10 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0061/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 62. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation11

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 11 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0062/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 63. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation12

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 12 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0063/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 64. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation13

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 13 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0064/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 65. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation14

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 14 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0065/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 66. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation15

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 15 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0066/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 67. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation16

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 16 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0067/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 68. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation17

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 17 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0068/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 69. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation18

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 18 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0069/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 70. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation19

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 19 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0070/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 71. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation20

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 20 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0071/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 72. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation21

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 21 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0072/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 73. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation22

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 22 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0073/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 74. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation23

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 23 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0074/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 75. Area of a Half-Open Unit-Cell Union

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

View answer and explanation24

Deduplicate coordinate pairs and count the distinct unit cells once each, giving 24 squared units. Repeated listings add no area, and an empty list has area zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0075/)

Exact area of half-open rectangles and unit-cell unions · qf-area-unit-cell-union · candidate





## 76. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 0) and B=[2, 2)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation0

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 0 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0076/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 77. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 2)×[0, 1) and B=[4, 6)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation8

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 8 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0077/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 78. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 2) and B=[3, 5)×[0, 2) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation10

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 10 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0078/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 79. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 3) and B=[1, 4)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation14

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 14 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0079/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 80. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[1, 3)×[1, 2) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation12

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0080/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 81. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 2)×[0, 3) and B=[0, 2)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation6

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 6 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0081/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 82. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 8) and B=[1, 4)×[2, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation6

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 6 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0082/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 83. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 2) and B=[5, 8)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation18

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 18 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0083/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 84. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[4, 7)×[0, 3) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation21

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 21 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0084/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 85. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 4) and B=[1, 5)×[1, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation23

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 23 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0085/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 86. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[1, 4)×[1, 3) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation20

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0086/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 87. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation12

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0087/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 88. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 9) and B=[1, 5)×[2, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation12

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0088/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 89. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 3) and B=[6, 10)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation32

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 32 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0089/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 90. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[5, 9)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation36

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 36 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0090/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 91. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 5) and B=[1, 6)×[1, 6) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation34

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 34 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0091/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 92. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 5) and B=[1, 5)×[1, 4) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation30

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0092/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 93. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 4)×[0, 5) and B=[0, 4)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation20

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0093/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 94. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 10) and B=[1, 6)×[2, 6) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation20

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 20 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0094/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 95. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 4) and B=[7, 12)×[0, 6) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation50

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 50 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0095/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 96. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 5) and B=[6, 11)×[0, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation55

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 55 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0096/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 97. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 6)×[0, 6) and B=[1, 7)×[1, 7) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation47

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 47 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0097/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 98. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 7)×[0, 6) and B=[1, 6)×[1, 5) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation42

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 42 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0098/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 99. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 5)×[0, 6) and B=[0, 5)×[0, 6) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation30

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0099/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate





## 100. Area of a Union of Two Half-Open Integer Rectangles

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 11) and B=[1, 7)×[2, 7) are half-open rectangles. Find area(A∪B) in squared length units.

View answer and explanation30

For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 30 squared units. Disjoint or merely touching rectangles have zero overlap area.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r06-area-0100/)

Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r06-area-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261005-advance-20261006-r06-area-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r06-area-foundations/validation-report.json)