Sector area in degrees:
formula and worked example

The pie slice and where it appears

A sector is the region enclosed by two radii and the arc between them: a slice of pizza, a fan-shaped sprinkler zone, the swept area of a windscreen wiper, a pie chart wedge, a curved lawn bed. The bounding angle at the centre decides how much of the whole circle it takes.

Measuring that angle in degrees is the most familiar approach, and it makes the geometry easy to picture: 90 degrees is a quarter, 180 a half, 45 an eighth. The formula below simply turns that fraction into an area.

Formula and inputs

Sector area equals pi x radius squared x angle / 360. The first part, pi x r^2, is the area of the full circle; the fraction angle/360 selects the share you want. If you are given the angle in radians, the same area is half of the radius squared times the angle, with no 360 and no pi.

Two details matter. The angle must be the central angle between the two radii, not an angle measured at the edge. And the radius must be the straight-line distance from the centre to the arc, not the length of the arc itself.

Everything here is proportion. A 30 degree slice is one twelfth, a 60 degree one is a sixth and a 120 degree one a third. Reducing the angle to a simple fraction first often makes the arithmetic easy to do mentally and gives a ready check on a calculator result.

A 45 degree sector with radius 7

Take a radius of 7 units and a central angle of 45 degrees.

1. 1Area of the full circle: 3.14159 x 7 x 7 = 153.938.
2. 2Fraction of the circle: 45 / 360 = 1/8 = 0.125.
3. 3Multiply: 153.938 x 0.125 = 19.242.
4. 4Quote the unit: 19.242 square units.

Because 45 degrees is exactly an eighth of a turn, you can check the answer by dividing the full circle area by eight. 153.938 / 8 = 19.242, so the working is consistent.

What the result means in practice

The area is the amount of surface swept or covered by the slice, so it is the number to use for seed, turf, paint, cover fabric or irrigation estimates. For a sprinkler rated to cover a 90 degree quarter with a throw of 7 m, the covered area is about 38.5 square metres, and you would divide a hose-timer budget accordingly.

Remember that a sector includes the triangle formed by the radii and the chord. If the shape you actually have is the sliver between the chord and the arc (a segment), you must subtract the triangle, which is a different calculation. A bowl-shaped curved bed is usually a segment; a fan-shaped one is a sector.

A pie chart is a familiar everyday use. The slice for a 25% share has an angle of 90 degrees, and its area is exactly a quarter of the disc regardless of the radius. The calculation shows what the eye already suspects, that area and angle move together.

Lengthening the radius at a fixed angle

The radius moves from 7 to 7.7 and the sector area from 19.242 to 23.283, a rise of 21%. Because the angle is the same, the sector is still an eighth of the circle; the circle itself is simply 21% larger. If you want to double a sector area while keeping the angle, the radius needs only to grow by about 41%, not 100%.

Conversely, at a fixed radius, area is directly proportional to angle: 90 degrees gives twice the 45 degree area, and 135 degrees gives three times. That linear behaviour in angle and quadratic behaviour in radius together explain most sector sizing surprises.

If a design brief fixes the area instead, the algebra reverses neatly. For a target area A at angle theta degrees, the radius needed is the square root of 360 x A divided by pi x theta, which tells you straight away how far a fan-shaped bed must reach.

Mistakes that turn up with sectors

• Leaving out the division by 360, which returns the whole circle's area times the number of degrees.
• Feeding a radian value into a degrees formula. An angle of 0.785 radians (which is 45 degrees) would be read as less than one degree.
• Using the diameter as the radius, an error that quadruples the answer.
• Mistaking a segment for a sector. The sector includes the triangle to the centre; the segment does not.

Scenario: a fan-shaped lawn bed

You are laying out a curved bed at a corner of a garden with a 4 m radius that fans out over 120 degrees. The area is pi x 16 x 120 / 360 = 16.76 square metres. With soil mulch to a depth of 7 cm, that is about 1.17 cubic metres. If the bed sits in a 90 degree corner the angle would be 90 and the area 12.57; checking both helps you decide whether the wider layout is worth the extra turf and edging on the curved side.

Seen from the planner's side, changing the angle is the cheaper lever. Opening the bed from 90 to 120 degrees adds a third to the area with the same radius and the same centre post, whereas increasing the radius by a third adds nearly 80%.