# Question 2720: Area of a Half-Open Integer-Endpoint Rectangle

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0004-02720 · q-20261005-advance-20261006-r06-area-0045



## Original question and givens

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.


- G001 — 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.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-area](https://ocme.evemisslab.com/en/mko/mko-area/)





## Step-by-step derivation

S001 · impliesBefore

orthonormal equal-length coordinate units;half-open regions

After

area in squared length units

Conditions: none

Axes are orthogonal with equal length units;half-open boundaries make adjacent unit cells disjoint.Area uses squared units,not length units.

Step identifiers and dependenciesRule: half_open_area_convention · Uses: G001





S002 · calculationBefore

coordinate rectangle width := sub(-19, -23)

After

coordinate rectangle width = 4

Conditions: none

Right minus left is the nonnegative width;translation of the origin contributes no area.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

coordinate rectangle height := sub(27, 7)

After

coordinate rectangle height = 20

Conditions: none

Top minus bottom is the nonnegative height,using endpoints of the same rectangle.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

coordinate rectangle area := mul(4, 20)

After

coordinate rectangle area = 80

Conditions: none

Rectangle area is width times height;a zero side makes the half-open rectangle empty,with area0.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

[-23, -19, 7, 27]

After

area=80

Conditions: none

Endpoints may coincide but may not be reversed;compute side lengths before multiplying,not products of coordinates.

Step identifiers and dependenciesRule: rectangle_area_boundary · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute coordinate rectangle area with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute coordinate rectangle height with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute coordinate rectangle width with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

80

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: d4c042793509f1ccba2fe2650ca2728d71f0f534f06ada19f15ceece1a9d0eb7
- candidate record SHA-256: 4188f203a8d7293d15c2363d977daca6dee375141696f5ccdda91df82bb53484
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r06-area-0045.json)