Question Video: Finding the Measure of a Central Angle Mathematics

Video Transcript

In the figure, given that the circumference is 96 and arc 𝐴𝐵 equals 12, find 𝜃.

So if we look at our diagram, we can add in the fact that the arc length 𝐴𝐵 is 12. We’re told that the circumference is 96. So that means the distance around the outside of the circle is 96. So as a fraction, the arc length over the circumference would be 12 over 96. And we can use this to find the fraction of 𝜃 out of the entire circle.

To do this, we need to know the sum of the angles in the circle. We can recall that the sum of the measures of the central angles in a circle is 360 degrees. You may also know this as the sum of the angles about a point is 360 degrees.

So to find the fraction that 𝜃 is out of the total angles in the circle, we write 𝜃 over 360 degrees. And we set this equal to the fraction of the arc length over the circumference. We now need to solve for 𝜃, and we can do this by finding the cross-product.

We can first notice, however, that we can simplify our fraction. Since 12 goes into both 12 and 96, we can write that 𝜃 over 360 degrees is equal to one over eight. Taking the cross-product then, we have 𝜃 times eight, which is eight 𝜃. And this is equal to 360 degrees times one, which is 360 degrees. To solve for 𝜃, we divide both sides of our equation by eight, giving us 360 degrees over eight. And since 360 divided by eight is 45, then we have 𝜃 equals 45 degrees.