Circles, arcs and sectors

Example: a circle has a diameter of 14 cm. What is its radius?

1. Radius = diameter ÷ 2
2. 7 cm

Circumference = πd = 2πr (on the formulae sheet)

π (pi) is about 3.14159… Use the π button on a calculator.

Example: a circle has a radius of 6 cm. Find its circumference, in terms of π and to 1 decimal place.

1. C = 2 × π × 6
2. 12π cm
3. 12π = 37.699…, so 37.7 cm to 1 d.p.

"In terms of π" means leave π in the answer, as an exact value: 12π, not 37.7.

Area = πr² (on the formulae sheet)

1. Find the radius (halve the diameter if you need to).
2. Square the radius, then multiply by π.

Example: a circle has a diameter of 10 cm. Find its area, in terms of π and to 1 decimal place.

1. Radius = 10 ÷ 2 = 5 cm
2. A = π × 5² = 25π cm²
3. 25π = 78.539…, so 78.5 cm² to 1 d.p.

An arc or a sector is a fraction of the whole circle: the angle at the centre over 360°.

• Arc length = θ/360 × 2πr
• Sector area = θ/360 × πr²

Learn the method: these aren't on the formulae sheet.

Example: a sector has a radius of 9 cm and an angle of 80°. Find its arc length, area and perimeter, to 1 decimal place.

1. Fraction: 80/360 = 2/9
2. Arc length = 2/9 × 2 × π × 9 = 4π = 12.6 cm
3. Area = 2/9 × π × 81 = 18π = 56.5 cm²
4. Perimeter = arc + 2 radii = 4π + 18 = 30.6 cm

A segment is a sector with a triangle taken away.

1. Find the area of the sector.
2. Find the area of the triangle made by the two radii and the chord.
3. Subtract.

Example: a quarter circle has a radius of 10 cm. A chord joins the ends of its two radii. Find the area of the segment, to 1 decimal place.

1. Sector: ¼ × π × 10² = 25π = 78.539…
2. The radii are at 90°, so the triangle is ½ × 10 × 10 = 50
3. 78.539… − 50 = 28.5 cm²

For a sector angle other than 90°, find the triangle's area with ½ab sin C (Higher tier).