You can find both the perimeter and the area of a shape. Here is a puzzle: if two rectangles have the same perimeter, must they have the same area?
| Rectangle | Perimeter | Area |
|---|---|---|
| 1 by 7 | 16 | 7 |
| 2 by 6 | 16 | 12 |
| 3 by 5 | 16 | 15 |
| 4 by 4 | 16 | 16 |
Why it works
The perimeter only fixes the total length around. With 16 units of fence, the length and width must add to 8 (half of 16), but they can split 8 in different ways: 1 and 7, 2 and 6, 3 and 5, 4 and 4. A long, thin rectangle encloses little space. As the sides get closer to equal, the area grows, and the square encloses the most.
Key idea
Rectangles with the same perimeter can have different areas, and rectangles with the same area can have different perimeters. Always work out the one you need.
Worked example: Same perimeter
A 2-by-6 rectangle and a 4-by-4 square both have a perimeter of 16. How much more area does the square have?
- Rectangle: $2 \times 6 = 12$.
- Square: $4 \times 4 = 16$.
- $16 - 12 = 4$.
Answer: 4 square units more
Worked example: Same area
A 2-by-6 and a 3-by-4 rectangle both have an area of 12. Compare their perimeters.
- $2 + 6 + 2 + 6 = 16$.
- $3 + 4 + 3 + 4 = 14$.
- The 2-by-6 rectangle's perimeter is 2 units longer.
Answer: 16 and 14
Worked example: From the perimeter
A rectangle has a perimeter of 20 cm and one side of 3 cm. What is its area?
- Length plus width is half the perimeter: $20 \div 2 = 10$.
- The other side is $10 - 3 = 7$ cm.
- Area: $3 \times 7 = 21$ square cm.
Answer: 21 square cm
Common mistake
Assuming the same perimeter means the same area. The table shows four rectangles, all with perimeter 16, whose areas are 7, 12, 15 and 16. Check by working out both measures for each shape.