Arc Length Formula

The arc length formula is used to find the length of an arc of a circle. An arc is a part of the circumference of a circle.

Again, when working with π, if we want an exact answer, we use π. If we want to approximate an answer, we substitute a rounded form of π, such as 3.14.Also, r refers to the radius of the circle which is the distance from the center to circumference of a circle. The symbol theta, θ, is used for angle degree measures.

Example 1:

Find the arc length of an arc formed by 60° of a circle with a radius of 8 inches.

Step 1:

Find the variables.

θ = 60°

r=8

Step 2:

Substitute into formula.

$Length=\frac{60°}{360°}2\pi \left(8\right)$

Step 3:

Evaluate for Arc Length

$Length=\frac{16\pi }{6}$

$Length=\frac{8\pi }{3}$

If you want an approximate answer, use 3.14

$Length=\frac{8\left(3.14\right)}{3}$

Length =8.37

Answer:

The length is about 8.37 inches.

Example 2:

Find the arc length of an arc formed by 75° of a circle with a diameter of 18cm.

Step 1:

Find the variables.

θ = 75°

r = 9 since that is half of the diameter.

Step 2:

Substitute into formula.

$Length=\frac{75°}{360°}2\pi \left(9\right)$

Step 3:

Evaluate for Arc Length

$Length=\frac{75\cdot 18\pi }{360}$

$Length=\frac{15\pi }{4}$

If you want an approximate answer, use 3.14

$Length=\frac{15\left(3.14\right)}{4}$

Length = 11.78

Answer:

The length is about 11.78 inches.