The length of a circular arc, also called arc length , can be defined as a fraction of a circle's circumference.
The distance around an entire circle is called the perimeter of the circle or circumference. However, the distance around a portion of a circle is called arc length.
The length of a circular arc is the product of the ratio (measure of the arc / 360) and the circumference of the circle (2πr)
Suppose θ is equal to 360 degrees, then the measure of arc AB is 360.
Length of arc AB = (360/360) × 2πr
Length of arc AB = 1 × 2πr = 2πr
Since 360 degrees is a full circle, it makes sense that 2πr is length of the full circle.
The formula above shows that if θ is a degree measure, then the length of arc AB = (θ/360) × 2πr
If θ is a radian measure, then the measure of the central angle is
(θ/2π radians) or simply θ/2π
Length of arc AB = (θ/2π) × 2πr
Length of arc AB = (θ/2π) × 2πr/1
Length of arc AB = (θ × 2πr)/(2π × 1)
Length of arc AB = (θ × 2πr)/2π
Length of arc AB = (θ × r)
A circle has a radius of 20 inches. Find the length of the arc intercepted by a central angle of π/4.
s = rθ = 20 × π/4 = 5π = 5(3.14) = 15.7 inches
You could also convert π/4 into degrees if you wish, but then you have to use the formula below.
Length of arc AB = (m arc AB / 360) × 2πr
π/4(180/π) = 180/4 = 45
Length of arc AB = (45/360) × 2πr
Length of arc AB = (0.125) × 2π(20)
Length of arc AB = (0.125) × 40π = 5(π) = 15.7 inches