Same Perimeter, Different Area (Grades 5–6)
Two shapes can have the same perimeter but very different areas.
Understanding same perimeter, different area
With a fixed amount of fence (perimeter), the area you enclose changes with the shape — and for a rectangle, the area is largest when it's closest to a square. With perimeter 12 (so length + width = 6): a 1×5 holds 5, a 2×4 holds 8, a 3×3 holds 9 — the square wins.
Seeing it in action
Worked example
Perimeter 12. Which rectangle holds the most area: 1×5, 2×4, or 3×3?
Areas: 1×5 = 5, 2×4 = 8, 3×3 = 9. → 3×3 (the square), area 9.
For the same perimeter, the squarer rectangle holds more area.
A common mistake, and the fix
Assuming equal perimeters mean equal areas.
A 1 by 5 and a 3 by 3 rectangle both have perimeter 12, but they hold 5 and 9 square units. Work out each area: for a fixed perimeter, the closer the rectangle is to a square, the more it holds.
Try a few
Perimeter 16: compare 1×7 vs 4×4 areas.
1 × 7 = 7 and 4 × 4 = 16; both have perimeter 16.
Perimeter 20: best rectangle area?
sides adding to 10, closest to a square.
Perimeter 8: 1×3 vs 2×2 areas?
1 × 3 = 3 and 2 × 2 = 4.
Perimeter Patrol
A calm tile-grid game for building and reading area and perimeter.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
Put the reasoning on paper.
Try six fraction, angle, perimeter and area questions with explained answers, for grades 4–6.
Free printable · 3 pagesText version (screen-reader friendly): Draw it. Explain it. Math sample
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