Question

What is the arc length of the subtending arc for an angle of 72 degrees on a circle of radius 4?

Answer 1

Let $L$ be the length of an arc subtended by an angle of $\theta$ on a circle of radius $r$. Then the ratio of the arc's length to its subtended angle is proportional with the circle's entire circumference to one complete revolution:

$\dfrac{2\pi r}{2\pi}=\dfrac{L}{\theta}$

Solving for $L$ yields the formula for arc length of a circular arc,

$L=r\theta$

where $\theta$ is in radians. To convert to degrees, use the conversion factor $\dfrac{180^\circ}{\pi\text{ rad}}$.

$L=4\times\left(72^\circ\times\dfrac{\pi\text{ rad}}{180^\circ}\right)=4\times\dfrac{2\pi}5=\dfrac{8\pi}5\approx5.0265$