Same Perimeter, Different Area (Grades 5–6)
Two shapes can have the same perimeter but very different areas. 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.
Understanding same perimeter, different area
Two shapes can have the same perimeter but very different areas. 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.
Key Idea
Two shapes can have the same perimeter but very different areas. 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.
Try a few
Perimeter 16: compare 1×7 vs 4×4 areas.
Perimeter 20: best rectangle area?
Perimeter 8: 1×3 vs 2×2 areas?
Perimeter Patrol
A calm tile-grid game for building and reading area and perimeter.
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