Arc Length Calculator

Central angle
Arc length
Sector area

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. In this article, we explain the arc length formula in detail and provide you with a step-by-step instruction of 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 area of a sector 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 radius of the circle. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.