📐 Same Area, Different Perimeter

What you’ll learn

1. Two Different QuestionsDefine area as the space inside a shape (counted in squares) and perimeter as the distance around its edge, and keep the two ideas separate.Area is the room inside a shape, measured by counting unit squares; for a rectangle it is length times width. Perimeter is the distance all the way around the edge — the fence. They answer two different questions about the same shape.
2. Building Shapes on a GridUse grid paper to measure a rectangle's area by counting inside squares and its perimeter by counting edge units.On a unit grid, a 2-by-6 rectangle has an area of 12 squares (2 rows of 6) and a perimeter of 16 units (6+6+2+2). Seeing both measured on the same shape sets up the key surprise.
3. Same Area, Different PerimeterShow that rectangles with the same area can have different perimeters.Twelve squares can be arranged as 1x12, 2x6, or 3x4 — all area 12 — yet their perimeters are 26, 16, and 14. Equal area does not mean equal perimeter; the shape matters.
4. The Squarest Shape Needs the Least FenceRecognise that, for a fixed area, the squarer (more compact) rectangle has the smaller perimeter.Stretching a fixed area into a long thin shape creates lots of edge and a large perimeter, while a compact square shape tucks squares inside and needs the least fence. Ranking 1x12, 2x6, 3x4 by perimeter makes this pattern clear.
5. Same Perimeter, Different AreaShow that rectangles with the same perimeter can enclose different areas, with the square enclosing the most.With a fixed 16 units of fence, a 1x7 encloses 7 squares, a 2x6 encloses 12, and a 4x4 square encloses 16. Equal perimeter does not mean equal area, and the square holds the most.
6. Why This Matters in Real LifeApply the area–perimeter relationship to real decisions such as fencing a field, building garden beds, and designing devices.Because a square encloses the most area for a fixed perimeter, a farmer fencing a fixed length gets the most grazing from a square pen. The same trade-off guides garden beds, room layouts, and even packing area onto chips and screens.
7. Prove It YourselfSummarise and test both truths: equal area can give different perimeters, and equal perimeter can give different areas, with squarer shapes being most efficient.The two big results — same area, different perimeter; same perimeter, different area — both follow from one rule: the squarer a rectangle, the less fence it needs and the more space it holds. Long thin shapes are wasteful both ways.

Questions this course answers

Grounded in trusted sources

• https://en.wikipedia.org/wiki/Area
• https://en.wikipedia.org/wiki/Perimeter
• https://en.wikipedia.org/wiki/Isoperimetric_inequality

Every Wunder lesson is built from real, reputable sources — never invented.

Related Math courses