VectorPaint equal-perimeter rectangles: check boundary length separately from area

English translation of the Chinese original, translated on 2026-11-08. Original publication dates, attribution, code and version notes are retained.

Equal boundary length does not imply equal enclosed area. Draw three rectangles of perimeter twenty units, checking four sides before counting unit squares instead of judging by appearance. Unlike an area-multiple study, this fixes perimeter while width and height vary together. All quantities are invented geometry, not automatic measurement or design optimisation.

Fix one unit scale and three side pairs

List A as width six, height four; B as eight by two; C as five by five, all in the same unit. At forty pixels per unit, A is 240 by 160 pixels, B 320 by 80 and C 200 by 200. Place titles outside rather than crowding interiors. Preserve this common scale instead of independently filling equal presentation boxes before comparing area. These are exercise geometry units, not automatically centimetres in a PNG; printing scale needs separate verification.

Calculate perimeter along all four sides

A’s boundary lengths are six,four,six,four, totalling twenty; B’s are eight,two,eight,two, also twenty; C has four sides of five, again twenty. Write two times the sum of width and height, not two times width plus height only once. Optional unit ticks belong on boundaries; internal grid lines do not contribute to perimeter. Hide area values initially and compare boundary calculations with geometry. Equal perimeter is the shared condition, not equal width, height or bounding boxes.

Represent area with equally sized unit squares

Add pale forty-pixel grids inside all three rectangles, each cell one square unit. A has six columns/four rows = 24 cells; B eight/two = 16; C five/five = 25. Keep counting grids from being obscured by heavy boundary strokes, confining them to interiors with a separate outer edge if needed. Label area in square units, distinct from perimeter units. Gridding only A while leaving B empty can suggest differing complexity rather than differing area, so use the same counting convention throughout.

Compare areas while keeping perimeter fixed

Read areas in ascending order: B sixteen, A twenty-four, C twenty-five, retaining perimeter twenty beside each. A exceeds B by eight square units, and C exceeds A by one; these are not side-length differences of eight or one. B is visibly wider but shorter and encloses less area, illustrating why shared scale and numbers matter. Colour may distinguish A/B/C without also encoding area through darkness and creating a second unexplained scale. Establish understanding through dimensions and cells before decoration.

Define the comparison set before saying largest

Among the three drawn cases, C is largest by direct comparison. For the additional claim that five by five maximises area among positive-integer-side rectangles of perimeter twenty, first list all unordered side pairs: one/nine, two/eight, three/seven, four/six, five/five, with areas 9,16,21,24,25. Swapping width/height changes orientation, not the combination. This finite list checks the stated integer scope; three pictures alone do not prove a theorem for arbitrary shapes. Circles, other polygons and three-dimensional volumes remain outside this conclusion.

Test condition preservation with a wrong-dimension copy

Duplicate B, changing width eight to seven while retaining height two, and clearly label it a counterexample. Its perimeter is eighteen and area fourteen, outside the fixed-twenty condition; updating only area under an equal-perimeter title is incorrect. Change width from 320 to 280 pixels and update its label, checking both before perimeter. Restore original B and verify area sixteen/perimeter twenty. The wrong copy identifies missed fields rather than adding a fourth valid case, and thicker outlines cannot restore two missing boundary units.

Check that output layout preserved the common scale

Save the source and inspect a viewing-size PNG for equal cell dimensions, then read side lengths, perimeter, area and units. If too small, enlarge everything uniformly or reduce text rather than enlarging B alone to make outer heights equal. If separate equal-size thumbnails are needed, label individual scales and state that visual area comparison is invalid there. Manually drawn grids are artwork objects and should export clearly when retained. Hide unwanted construction lines actively; their names do not define export behaviour.

Troubleshoot boundary and interior quantities separately

For perimeter mismatch, inspect four sides and dimension labels. For area mismatch, count columns times rows rather than forcing both quantities with one correction. If areas look equal despite different numbers, inspect independent scaling. Repair square-unit wording without changing geometry. Preserve original dimensions, boundary calculations, area list and counterexample, tracing every number to its definition. Completion means equal-perimeter/different-area relationships agree in words and graphics, not automatic optimal-layout verification or real land/packaging-material estimates.