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 r}=\frac{x}{360}, where L is arc length and x is a central angle, so x=\frac{180L}{\pi 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 \cdot 70}{\pi \cdot 100} x=\frac{180 \cdot 90}{\pi \cdot 100}
x\approx 40.1 x\approx 51.6

So x measures between 40.1 and 51.6 degrees.

This problem had to do with an angle measured in degrees. What if you’re asked about an angle measured in radians?

Mini-lesson: What’s a radian?

Radian is a unit of angle measure. Think of a circle. Imagine you cut a string equal to the circle’s radius and then lay the string out on the circle. The arc covered by the string is one radian. The central angle that corresponds to that arc is also one radian. 

One radian is the measure of a central angle whose sides intersect an arc that is as long as the circle’s radius.

In other words, one radian is 1/(2π) of the circle’s circumference. A whole circle’s arc is 2π radians — but a whole circle is also 360°. So 2π radians = 360°.

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.