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Math & Statisticsgeometry

Arc Length and Sector Area Calculator

Arc length, sector area, chord and segment from a radius and an angle. The formulas are trivial in radians and awkward in degrees, which is the whole argument for radians.

Also called: sector area calculator, chord length.

Angle in
Arc length
10.47

Arc length 10.47 on a radius of 10. The sector area is 52.36 and the straight chord across it is 10.

Sector area
52.36
Chord length
10
Segment area
9.06
Angle in radians
1.05
How this is calculated
Method and background

How this is calculated

Arc length is simply radius times angle when the angle is in radians, because a radian is defined as the angle whose arc equals the radius. In degrees the same formula needs a conversion factor. The chord is the straight line across the arc, and the segment is the area between the chord and the arc, which is the sector less the triangle.

arc length = radius * angle in radians, which is why radians are the natural unit
r
Radius
angle
The angle, in radians

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

sixty degrees on a radius of ten

Radius
10
Angle
60
Angle in
Degrees

Arc length10.47

At 60 degrees the chord equals the radius, which is the equilateral case

Open this example

a full turn is the circumference

Radius
10
Angle
360
Angle in
Degrees

Arc length62.83

boundary: 2 pi r

Open this example

Method and limits

What it assumes

  • A circular arc.

What it deliberately does not model

  • Angles above a full turn wrap in practice, though the arithmetic continues.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why are radians easier here?
Because arc length is exactly radius times angle in radians, with no constant. That is the definition of a radian, and it is why every trigonometric identity in calculus assumes them.