# Arc Length and Central Angle

## Problem

Pictured is a circle with central angle x, radius 100, and an arc whose length is somewhere between 70 and 90.  What are the possible values, in degrees, of x?

## Solution

An arc is a fraction of a circumference. What fraction? An arc’s length is the same proportion of a circle’s circumference as its central angle is of a whole circle. And a circle’s circumference equals 2π times the circle’s radius.

Then $\frac{L}{2\pi&space;r}=\frac{x}{360}$, where L is arc length and x is a central angle, so $x=\frac{180L}{\pi&space;r}$.

Consider the extremes of the range you’re given: $L=70$ and $L=90.$ What is x for those values of arc length?

 For L=70: For L=90: $x=\frac{180&space;\cdot&space;70}{\pi&space;\cdot&space;100}$$x=\frac{180&space;\cdot&space;70}{\pi&space;\cdot&space;100}$ $x=\frac{180&space;\cdot&space;90}{\pi&space;\cdot&space;100}$$x=\frac{180&space;\cdot&space;90}{\pi&space;\cdot&space;100}$ $x\approx&space;40.1$$x\approx&space;40.1$ $x\approx&space;51.6$$x\approx&space;51.6$

So x measures between 40.1 and 51.6 degrees.

For an angle $\theta$ measured in radians, the length of a circle’s arc, L,  equals the central angle times the radius. $L=r\theta$.