Sector Area Calculator

What this calculator does

Picture a slice of pizza, or a single wedge cut from a round cake. This tool measures the whole slice for you, three ways at once: the area inside it, the length of the arc (the curved crust along the outside), and the length of the chord (the straight cut across the open end).

Give it the angle of the slice and the radius of the circle it came from, and it hands back all three.

Using the calculator

1. Type the central angle, the angle at the pointed tip of the slice, and choose degrees or radians.
2. Type the radius.
3. Press Calculate.

You get the sector area, the arc length, and the chord length together. The radius has no unit attached here, so pick whatever unit you like and read the results in it: the area in that unit squared, the arc and chord in that same unit.

How the three measurements are worked out

All three come from the same slice, so they share one idea: a sector is a fraction of the whole circle, set by its angle.

The sector area is that fraction of the circle's area:

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

The arc length is the same fraction of the circle's circumference:

arc length = (2 × π × radius × angle) ÷ 360

The chord is different. It is the straight line joining the two ends of the arc, and it drops out of the triangle made by the two radii:

chord = 2 × radius × sin(angle ÷ 2)

If you enter the angle in radians, the tool converts it to degrees first, so the formulas above still hold. In radians the first two shorten to the familiar area = ½ × radius² × angle and arc length = radius × angle.

The curved edge and the straight edge

The arc and the chord are easy to confuse, so it is worth being clear. The arc is the curved edge, the long way round the rim of the slice. The chord is the straight shortcut between its two corners.

For the same slice the arc is always the longer of the two, and they only meet when the slice shrinks down to nothing. On a 60 degree slice, for instance, the curved crust is clearly longer than the straight cut across it. If you only need the area of the slice in real units like cm², the area of sector calculator is the simpler tool for that.

Units, angles, and rounding

There are no unit menus here on purpose, so the tool stays unit-agnostic. Keep your radius in one unit and read everything in it: a radius in metres gives an area in square metres and an arc and chord in metres. Every result is rounded to three decimal places.

The angle is yours to set. If you push it past 360 degrees the slice has wrapped right around the circle, so for a single real wedge keep it at 360 or below.

A worked example | 60 degrees on a radius of 10

Say the slice has an angle of 60 degrees on a radius of 10.

• Sector area: (π × 10² × 60) ÷ 360 = (π × 100) ÷ 6 = 52.360 square units.
• Arc length: (2 × π × 10 × 60) ÷ 360 = (2 × π × 10) ÷ 6 = 10.472 units.
• Chord: 2 × 10 × sin(30 degrees) = 20 × 0.5 = 10.000 units.

So a 60 degree slice of a radius-10 circle holds an area of about 52.36, with a curved edge of 10.47 and a straight edge of exactly 10. The arc is longer than the chord, just as it should be.

Questions people ask

What is the difference between the arc and the chord?

The arc is the curved edge of the slice. The chord is the straight line across its open end, between the two corners. The arc is always the longer one.

Can I enter the angle in radians?

Yes. Pick radians from the dropdown and the tool converts it for you before working out the three measurements.

What is a sector of a circle?

The slice of a circle between two radii and the arc that joins them, just like a slice of pie.

Can the angle be more than 360 degrees?

The tool lets you, but past 360 the slice has gone all the way round the circle, so a normal single slice sits at 360 or below.

Why is there no unit on the radius?

So you can use any unit you like. Whatever you put the radius in, the area comes out in that unit squared and the arc and chord come out in that same unit.

References

A note on where these come from. The area and arc formulas are the circle's own area and circumference, each scaled by the fraction of the circle the angle covers, and both rest on the value of π as tabulated by the US National Institute of Standards and Technology. The chord formula, chord = 2 × radius × sin(angle ÷ 2), comes from the isosceles triangle made by the two radii, a standard result in plane trigonometry. For further reading, see Circular sector.

1. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/
2. The chord identity, chord = 2r · sin(θ ÷ 2), from the triangle formed by two radii (plane trigonometry).