Central Angles

Among the most important applications of the circle to human progress has been circle-based (disk-shaped) food items. This is a family of foods that includes, among other things, cookies, bagels, doughnuts, pies, and pizzas. (Some consider pizzas to be a subset of pies, but pepperoni pie doesn't sound nearly as delicious as pepperoni pizza.)

Critical to the development of circle-based foods was the concept of the central angle. Given ⊙O, an angle is a central angle of ⊙O if its vertex is at O. Simple as that.

Here, ∠1 is a central angle of ⊙O. So is ∠2. Every central angle has a buddy. The measure of a central angle and its buddy angle add up to 360°, the number of degrees in a full circle. In other words, buddy angles complete each other. Aww.

With the discovery of the central angle, people could easily share circle-based foods as they saw fit. If you wanted to divide a pizza among five people, you could cut slices based on central angles of 360° ÷ 5 = 72°.

Sample Problem

It's your birthday and you'll cry if you want to. There's no reason to cry though, since you got a massive chocolatey birthday cake. If there are 20 people total at your party, at what central angle should you cut the cake?

Regardless of how big the cake is, it has a central angle of 360° because it's a circle. If there are 20 people, we should cut everyone a slice that is 360° ÷ 20 = 18° in measure. Time to bust out the protractor.

Sample Problem

All circles are similar. True or false?

It's been a while since we talked about similarity, so here's a quick refresher: similarity exists when two figures are the same shape (all their angles are equal), but not the same size. This also means they can be carried onto each other using similarity transformations (translation, reflection, rotation, and dilation).

Circles have 360° total. That won't change ever, so we took care of the angle requirement (as well as the rotation and reflection requirements). If a single point and a length defines a circle, we can always translate the point to a different location and dilate the length so it matches another.

In other words, all circles are similar to each other because any similarity transformation can move one onto the other. In fact, only dilation and translation are needed, so we leave reflection and rotation at home.