Sector of a circle with the arc length marked
Central angle (θ)
deg
Radius (r)
m
Diameter
m
Sector area (A)
Chord length (c)
m
Arc length
Arc length (L)
m

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

Arc length formula

arc length

The length of an arc depends on the radius of a circle and the central angle θ. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:

L / θ = C / 2π

As circumference C = 2πr,

L / θ = 2πr / 2π
L / θ = r

We find out the arc length formula when multiplying this equation by θ:

L = r * θ

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

Area of a sector of a circle

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,

A / θ = πr² / 2π
A / θ = r² / 2

The formula for the area of a sector is:

A = r² * θ / 2

How to find the length of an arc and sector area: an example

  1. Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)
  2. What will be the angle between the ends of the arc? Let's say it is equal to 45 degrees, or π/4.
  3. Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 = 11.78 cm.
  4. Calculate the area of a sector: A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm².
  5. You can also use the arc length calculator to find the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

Make sure to check out the equation of a circle calculator, too!

FAQ

How do you find arc length without the radius?

To calculate arc length without radius, you need the central angle and the sector area:

  1. Multiply the area by 2 and divide the result by the central angle in radians.
  2. Find the square root of this division.
  3. Multiply this root by the central angle again to get the arc length.
  4. The units will be the square root of the sector area units.

Or the central angle and the chord length:

  1. Divide the central angle in radians by 2 and perform the sine function on it.
  2. Divide the chord length by double the result of step 1. This calculation gives you the radius.
  3. Multiply the radius by the central angle to get the arc length.

How do you find arc length using radians?

  1. Multiply the central angle in radians by the circle’s radius.
  2. That’s it! The result is simply this multiplication.

How do you calculate arc length without the angle?

To calculate arc length without the angle, you need the radius and the sector area:

  1. Multiply the area by 2.
  2. Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians.

Or you can use the radius and chord length:
  1. Divide the chord length by double the radius.
  2. Find the inverse sine of the result.
  3. Double the result of the inverse sine to get the central angle in radians.
  4. Once you have the central angle in radians, multiply it by the radius to get the arc length.

Does arc length have to be in radians?

Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

Bogna Haponiuk