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 explanation
0
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
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 explanation
2
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
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 explanation
6
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
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 explanation
12
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
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 explanation
20
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
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 explanation
5
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
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 explanation
12
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
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 explanation
21
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
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 explanation
32
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
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 explanation
45
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
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 explanation
10
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
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 explanation
22
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
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 explanation
36
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
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 explanation
52
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
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 explanation
70
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
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 explanation
15
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
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 explanation
32
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
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 explanation
51
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
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 explanation
72
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
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 explanation
95
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
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 explanation
20
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
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 explanation
42
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
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 explanation
66
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
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 explanation
92
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
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 explanation
120
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
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 explanation
0
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
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 explanation
4
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
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 explanation
9
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
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 explanation
16
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
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 explanation
5
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
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 explanation
12
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
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 explanation
21
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
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 explanation
32
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
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 explanation
9
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
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 explanation
20
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
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 explanation
33
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
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 explanation
48
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
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 explanation
13
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
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 explanation
28
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
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 explanation
45
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
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 explanation
64
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
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 explanation
17
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
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 explanation
36
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
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 explanation
57
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
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 explanation
80
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
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 explanation
21
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
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 explanation
44
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
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 explanation
69
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
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 explanation
96
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
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 explanation
25
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
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 explanation
0
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
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 explanation
1
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
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 explanation
2
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
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 explanation
3
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
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 explanation
4
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
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 explanation
5
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
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 explanation
6
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
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 explanation
7
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
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 explanation
8
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
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 explanation
9
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
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 explanation
10
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
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 explanation
11
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
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 explanation
12
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
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 explanation
13
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
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 explanation
14
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
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 explanation
15
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
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 explanation
16
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
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 explanation
17
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
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 explanation
18
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
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 explanation
19
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
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 explanation
20
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
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 explanation
21
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
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 explanation
22
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
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 explanation
23
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
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 explanation
24
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
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 explanation
0
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
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 explanation
8
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
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 explanation
10
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
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 explanation
14
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
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 explanation
12
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
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 explanation
6
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
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 explanation
6
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
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 explanation
18
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
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 explanation
21
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
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 explanation
23
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
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 explanation
20
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
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 explanation
12
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
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 explanation
12
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
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 explanation
32
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
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 explanation
36
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
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 explanation
34
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
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 explanation
30
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
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 explanation
20
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
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 explanation
20
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
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 explanation
50
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
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 explanation
55
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
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 explanation
47
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
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 explanation
42
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
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 explanation
30
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
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 explanation
30
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
Exact area of half-open rectangles and unit-cell unions · qf-area-two-rectangle-union · candidate
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