Area Of Sector Calculator

What this calculator does

A sector is a wedge of a circle, the shape of a slice of pie or a hand-held fan. This tool gives you the area inside that wedge, in real square units.

Tell it the radius of the circle and the angle of the slice in degrees, and it returns the area, labelled in square millimetres, square centimetres, and so on, to match whatever unit you measured in.

Using the calculator

1. Type the radius and pick its unit.
2. Type the central angle in degrees, up to 360.
3. Press Calculate.

The area comes back in square units of the radius's unit. The wider the angle, the bigger the slice, and the bigger the area.

How the sector area is worked out

A sector is just a fraction of the whole circle, and the angle tells you how big a fraction.

sector area = (π × radius² × angle) ÷ 360

Read it from the inside out. π × radius² is the area of the whole circle. A full circle goes all the way round, 360 degrees. Your slice covers angle degrees out of that 360, so it is angle ÷ 360 of the whole pie. Multiply the circle's area by that fraction and you have the slice.

This makes some cases easy to picture. A 90 degree slice is 90 ÷ 360, a quarter of the circle. A 180 degree slice is a half. A 36 degree slice is a tidy one tenth.

Working in real square units

The radius carries a unit, so the area lands in the matching square unit, since you are multiplying a length by a length: a radius in centimetres gives an area in square centimetres. The angle, being a share of the circle, is just a plain number and adds no unit of its own.

Switch the radius unit and the tool converts the radius for you, so it keeps pointing at the same real length. A tidy result is shown in full, while a long decimal is rounded to four places to keep it readable.

A worked example | a 90 degree slice on a 6 cm radius

Say the slice has a 90 degree angle on a 6 cm radius.

1. Work out the fraction of the circle: 90 ÷ 360 = one quarter.
2. Find the whole circle's area: π × 6² = π × 36 = 113.097 cm².
3. Take a quarter of it: 113.097 ÷ 4 = 28.2743 cm².

So a quarter slice of a 6 cm circle has an area of about 28.27 cm².

Why the angle stops at 360 degrees

The angle box stops at 360 degrees, and that is by design rather than a limit to work around. A sector is a piece of one circle, and a full 360 degree sector is already the entire circle. There is no slice bigger than the whole pie, so 360 is the natural ceiling. At exactly 360 the formula gives π × radius², the plain area of the circle, which is just what you would want.

If you also want the curved edge and the straight chord of the slice, or the option to work in radians, the sector area calculator gives all three together.

Questions people ask

What is the area of a sector with a 90 degree angle?

A quarter of the circle's area, since 90 is a quarter of 360. For a radius r that is (π × r²) ÷ 4.

What is a sector of a circle?

The wedge between two radii and the arc that joins them, like a slice of pie or a slice of pizza.

Is a 90 degree sector a quarter circle?

Yes. 90 out of 360 degrees is one quarter, so its area is a quarter of the whole circle's area.

Does this take radians?

This tool takes degrees. To turn radians into degrees, multiply by 180 and divide by π.

Why is the answer in square units?

Because area is a length times a length. A radius in cm gives an area in cm². The angle adds no unit, since it is only a fraction of the circle.

References

A note on where this comes from. The sector area is simply the area of the whole circle, π × radius², taken in the fraction the angle covers. That a circle's area is π × radius² was proved by Archimedes more than two thousand years ago in his Measurement of a Circle, and the value of π behind it follows the figure tabulated by the US National Institute of Standards and Technology. For further reading, see Circular sector.

1. Archimedes, Measurement of a Circle (c. 250 BCE), the proof that a circle's area equals π × radius².
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/