How to : 2 Different Ways to Find a Circle’s Arc Length

Set up the formula for arc length. The formula is ${displaystyle {text{arc length}}=2pi (r)({frac {theta }{360}})}$, where ${displaystyle r}$ equals the radius of the circle and ${displaystyle theta }$ equals the measurement of the arc’s central angle, in degrees.^[1]

Plug the length of the circle’s radius into the formula. This information is typically given to you in a problem. Otherwise, measure the circle’s radius with a ruler or protractor. Simply substitute the radius’ value for the variable ${displaystyle r}$.

• For example, if the circle’s radius is 10 cm, set up the formula like this: ${displaystyle {text{arc length}}=2pi (10)({frac {theta }{360}})}$.

Insert the value of the arc’s central angle into the formula. Typically, the problem you’re working on provides this information. Make sure to convert the angle to degrees if it’s currently in radians. Then, substitute the central angle’s measurement for ${displaystyle theta }$ in the formula.

• For example, if the arc’s central angle is 135 degrees, your formula now looks like: ${displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})}$.

Multiply the radius by $$2π{displaystyle 2pi }. If you are not using a calculator, use the approximation ${displaystyle pi =3.14}$ for your calculations. Rewrite the formula using this new value, which represents the circle’s circumference.^[2]

Divide the arc’s central angle by 360 degrees. Since a circle has 360 degrees total, dividing the central angle by 360 degrees gives you the portion of the circle that the sector represents. Using this information, find what portion of the circumference the arc length represents.

• For example, simplify the formula to get:
  ${displaystyle {text{arc length}}=(62.8)({frac {135}{360}})}$
  ${displaystyle =(62.8)(.375)}$

Multiply the two numbers together. This gives you the length of the arc.

• Solve the formula:
  ${displaystyle {text{arc length}}=(62.8)(.375)=23.55}$
  So, the arc length of a circle with a radius of 10 cm and a central angle of 135 degrees is about 23.55 cm.