A slice of cake, a pizza wedge, the area a windscreen wiper sweeps or a sprinkler that only waters a quarter circle: all of these are sectors. This article shows how to calculate their area and arc length, what radians are and how a sector differs from a segment.
What is a sector?
A sector is bounded by two radii and the arc between them. Its tip lies at the centre. Its size is set by the central angle α: at 90° it is a quarter circle, at 180° a semicircle.
The formulas with the angle in degrees
Because the sector is the fraction α ÷ 360° of the whole circle, you simply take that fraction of the circumference and the area:
Pizza example: a 30 cm pizza is cut into 8 slices. Each slice has α = 45° and r = 15 cm. The crust edge of one slice is 2 × π × 15 ÷ 8 = 11.78 cm long and its area π × 225 ÷ 8 = 88.36 cm².
Cake example: 12 slices from a 26 cm tin give each slice 30° and r = 13 cm – a 6.81 cm edge and 44.24 cm² of cake.
Garden example: a rotary sprinkler in the corner of the lawn is set to a quarter circle with an 8 m reach. It waters π × 64 ÷ 4 = 50.27 m².
Radians
Instead of degrees you can measure an angle as a length: radians give the length of the arc on a circle with radius 1. A full circle has circumference 2π, so 360° equals exactly 2π. To convert:
| Degrees | Radians exact | Radians decimal |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 360° | 2π | 6.2832 |
An angle of 1 radian (1 rad) equals 180° ÷ π = 57.30°. At that angle the arc is exactly as long as the radius.
The formulas in radians
In radians the formulas become particularly simple, because the factor 2π ÷ 360° disappears:
The last form is handy: if you know the arc length and the radius, you do not need the angle at all. The area is then like a triangle's “half base times height”, except that the base is curved.
Sector or segment?
The two are easily confused. A sector reaches all the way to the centre (a slice of cake). A segment is cut off by a chord (an arch). Its area is the sector minus the triangle formed by the two radii and the chord:
Example: r = 10 cm, α = 60°. The sector has 52.36 cm², the triangle 100 ÷ 2 × sin 60° = 43.30 cm². So the segment is only 9.06 cm².
Application: pie charts
Every pie chart is made of sectors. The angle for a share is percentage × 3.6°: 25% is 90°, 10% is 36°. The slice's area is then exactly that percentage of the circle's area.
In the circle calculator choose “Sector”, enter radius and angle and get arc length, area, chord and the angle in radians with a diagram. All circle formulas are in the guide Circle formulas.