Slice it and slide it

The idea

A circle of radius r has area A = πr² = (1/2) × circumference × radius; its slices rearrange into nearly a parallelogram with base πr and height r.

A circle in slices

Nīlakaṇṭha Somayājī (c. 1500) found the most visual way to see the area of a circle: cut the circle into slices and rearrange them. A circle of radius r has circumference 2πr.

Cut a circle into thin slices and lay them side by side, point up, point down, point up. What shape do they begin to make?

• A triangle
• Almost a parallelogram
• Another circle

Cut and rearrange

Pick the number of slices. Watch the bumpy edges flatten as the slices get thinner. Try all five numbers.

A parallelogram of base πr

A circle of radius r has area A = πr² = half the circumference × radius. Its slices rearrange into nearly a parallelogram with base πr (half the circumference) and height r.

Half the slices point up and half point down, so the base is half the circumference: 1/2 × 2πr = πr. The height is the radius r. Area of a parallelogram = base × height = πr × r = πr².

Notes

A circle of radius r has area πr² = half the circumference × radius: its slices make nearly a parallelogram of base πr and height r.

Check yourself

A circle has radius 7 cm. Use π ≈ 22/7 here. What is its area?

Answer: 154 cm²

A = 22/7 × 7 × 7 = 22 × 7 = 154 cm².

A circle has radius 10 m. Use π = 3.14. What is its area?

Answer: 314 m²

A = 3.14 × 10 × 10 = 314 m².

A circle is cut into slices that form nearly a parallelogram. What is the base of this parallelogram?

A circular pond has circumference 62.8 m. Use π = 3.14. What is the area of the pond?

Answer: 314 m²

r = 62.8 ÷ 6.28 = 10 m. Then A = 1/2 × 62.8 × 10 = 314 m².

• πr, half the circumference — correct. Yes. Half the arcs point up and half point down, so the base is half of 2πr.
• 2πr, the whole circumference. Only half of the arcs lie along the base. The other half lie along the top.
• r, the radius. The radius is the height of the parallelogram, not its base.