Scale it up, the ratio stays
Perimeter² to area ratio depends on shape
Scaling a shape
Early societies needed the area of a circle: to plan how much grain a round tower can hold, how big a round garden is, or how much tax to charge. They already knew that the area A must be proportional to the square of the circumference C. Why? Let us look at how P² : A behaves, where P is the perimeter.
A square has side 3, so P = 12 and A = 9. Now double the side to 6. What happens to the ratio P² : A?
What this lesson covers
The idea
For a given shape, the ratio P² : A of the square of its perimeter to its area stays fixed as its scale changes, depending only on the shape; so for circles C² : A is a constant.
Scaling a shape
Early societies needed the area of a circle: to plan how much grain a round tower can hold, how big a round garden is, or how much tax to charge. They already knew that the area A must be proportional to the square of the circumference C. Why? Let us look at how P² : A behaves, where P is the perimeter.
A square has side 3, so P = 12 and A = 9. Now double the side to 6. What happens to the ratio P² : A?
- It stays the same
- It doubles
- It becomes 4 times as big
Change the size
Pick each shape and change its size with + and −. Watch the last row of the table. Try two sizes of every shape.
The ratio belongs to the shape
For a given shape, the ratio P² : A of the square of the perimeter to the area stays fixed when the shape is scaled up or down. It depends only on the shape, not on the size. For circles, C² : A is a constant.
For a square of side a: P = 4a and A = a², so P² : A = 16a² : a² = 16 : 1 for every square. For an equilateral triangle of side a: P = 3a and A = (√3/4)a², so P² : A = 9a² : (√3/4)a² = 36 : √3. So for a circle, C² : A must also be some fixed number. The Babylonians measured it long before 1500 BCE and found it close to 12 : 1, so A ≈ C² ÷ 12.
Notes
For a given shape, P² : A stays fixed as the shape is scaled. It depends only on the shape: 16 : 1 for squares, 36 : √3 for equilateral triangles, and a constant for circles too.
Check yourself
A square has side 5. What is P² ÷ A?
Answer: 16
P² = 400 and A = 25, so P² ÷ A = 400 ÷ 25 = 16. It is 16 for every square.
For every square, P² : A = 16 : 1. A square has perimeter 20. Use the ratio to find its area.
Answer: 25
P² = 400, so A = 400 ÷ 16 = 25. This is a square of side 5.
What does the ratio P² : A depend on?
The Babylonians used C² : A ≈ 12 : 1 for circles, that is A ≈ C² ÷ 12. A circle has circumference 24. What area do they get?
Answer: 48
A ≈ 576 ÷ 12 = 48.
- On the size of the shape. It does not. A square with side 1 and a square with side 100 both give 16 : 1.
- Only on the shape, not on the size — correct. Yes. All squares have 16 : 1, however large or small.
- Nothing: it is the same for all shapes. Squares give 16 : 1 but equilateral triangles give 36 : √3. Different shapes give different ratios.