A share of the area

The idea

A sector whose arc subtends θ° at the centre of a circle of radius r has area πr² × θ°/360°; by symmetry a semicircular disc has area (1/2)πr² and a quarter disc (1/4)πr².

Half and quarter

The area of a sector is found the same way as the length of an arc. Reflect a circle in a diameter and the two halves are equal. Turn it by a quarter-turn and the four quarters are equal.

A circle of radius r has area πr². What is the area of a semicircular disc?

• Half of πr²
• The same, πr²
• Twice πr²

Turn the angle

Turn the angle θ at the centre. The bar shows what part of the whole circle the blue sector is.

A share of πr²

A sector whose arc subtends θ° at the centre of a circle of radius r has area πr² × θ°/360°. By symmetry a semicircular disc has area half of πr² and a quarter disc a quarter of πr².

A sector is θ/360 of the whole circle, so its area is θ/360 of πr². A semicircular disc has θ = 180 and 180/360 = 1/2. A quarter disc has θ = 90 and 90/360 = 1/4.

Notes

A sector of angle θ° in a circle of radius r has area πr² × θ°/360°. A semicircular disc has area half of πr² and a quarter disc a quarter of πr².

Check yourself

A circle has radius 10 cm. Use π = 3.14. What is the area of a semicircular disc of this circle?

Answer: 157 cm²

Circle = 314 cm², so the semicircular disc = 314 ÷ 2 = 157 cm².

A sector has an arc that subtends 60° at the centre of a circle of radius 6 cm. Use π = 3.14. What is the area of the sector?

Answer: 18.84 cm²

Area = 113.04 × 60/360 = 113.04 × 1/6 = 18.84 cm².

A sector has an angle of 120° at the centre of a circle of radius 21 cm. Use π ≈ 22/7 here. What is the area of the sector?

Answer: 462 cm²

πr² = 22/7 × 441 = 22 × 63 = 1386. Then 1386 × 1/3 = 462 cm².

A sector of a circle of radius 6 cm has area 28.26 cm². Use π = 3.14. What angle does it subtend at the centre? (in degrees)

Answer: 90 °

28.26 ÷ 113.04 = 1/4, so θ = 1/4 of 360° = 90°. It is a quarter disc.