Imagine you are standing in the middle of a perfectly circular arena. You look out toward the crowds along the sides of the arena, and you spot your best friend in one seat and your middle school math teacher a couple sections over. What's the distance between them and you? How far would you have to walk to travel from your friend's seat to your teacher's seat? What are the measures of the angles between you? These are all questions related to central angles.
A central angle is the angle that forms when two radii are drawn from the center of the circle to its edges. In this example, the two radii are your two lines of sight from you, at the center of the arena, to your friend, and your line of sight to your teacher. The angle that forms between these two lines is the central angle. It's the angle closest to the center of the circle.
Your friend and your teacher are seated along the circumference or the edges of the circle. The pathway along the arena that connects them is an arc.
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Find the Central Angle from Arc Length and Circumference
There are a couple of equations you can use to find the central angle. Sometimes you'll get the arc length, the distance along the circumference between two points. (In the example, this is the distance you would have to walk around the arena to get from your friend to your teacher.) The relationship between central angle and arc length is:
(arc length) ÷ circumference = (central angle) ÷ 360°
The central angle will be in degrees.
This formula makes sense, if you think about it. The length of the arc out of the total length around the circle (circumference) is the same proportion as the arc's angle out of the total angle in a circle (360 degrees).
To use this equation effectively, you need to know the circumference of the circle. But you can also use this formula to find the arc length if you know the central angle and the circumference. Or, if you have the arc length and the central angle, you can find the circumference!
Find the Central Angle from the Arc Length and Radius
You can also use the radius of the circle and the arc length to find the central angle. Call the measure of the central angle θ. Then:
θ = s ÷ r, where s is the arc length and r is the radius. θ is measured in radians.
Again, you can rearrange this equation depending on the information you have. You can find the length of the arc from the radius and the central angle. Or you can find the radius if you have the central angle and the arc length.
If you want the arc length, the equation looks like this:
s = θ * r, where s is the arc length, r is the radius, and θ is the central angle in radians.
The Central Angle Theorem
Let's add a twist to your example where you're in the arena with your neighbor and your teacher. Now there's a third person you know in the arena: your next-door neighbor. And one more thing: They're behind you. You have to turn around to see them.
Your neighbor is approximately across the arena from your friend and your teacher. From your neighbor's point of view, there's an angle formed by their line of sight to the friend and their line of sight to the teacher. That's called an inscribed angle. An inscribed angle is an angle formed by three points along a circle's circumference.
The Central Angle Theorem explains the relationship between the size of the central angle, formed by you, and the inscribed angle, formed by your neighbor. The Central Angle Theorem states that the central angle is twice the inscribed angle. (This assumes that you're using the same endpoints. You're both looking at the teacher and the friend, not anybody else).
Here's another way to write it. Let's call your friend's seat A, your teacher's seat B and your neighbor's seat C. You, at the center, can be O.
So, for three points A, B and C along the circumference of a circle and point O at the center, the central angle ∠AOC is twice the inscribed angle ∠ABC.
That is, ∠AOC = 2∠ABC.
This makes some sense. You're closer to the friend and the teacher, so to you they look further apart (a larger angle). To your neighbor on the other side of the stadium, they look much closer together (a smaller angle).
Exception to the Central Angle Theorem
Now, let's shift things up. Your neighbor on the far side of the arena starts to move around! They still have a line of sight to the friend and the teacher, but the lines and angles keep shifting as the neighbor moves. Guess what: As long as the neighbor stays outside the arc between the friend and the neighbor, the Central Angle Theorem still holds true!
But what happens when the neighbor moves between the friend and the teacher? Now your neighbor is inside the minor arc, the relatively small distance between the friend and the teacher compared to the larger distance around the rest of the arena. Then you reach an exception to the Central Angle Theorem.
The exception to the Central Angle Theorem states that when point C, the neighbor, is inside the minor arc, the inscribed angle is the supplement of half the central angle. (Remember that an angle and its supplement add to 180 degrees.)
So: inscribed angle = 180 - (central angle ÷ 2)
Or: ∠ABC = 180 - (∠AOC ÷ 2)
Math Open Reference has a tool to visualize the Central Angle Theorem and its exception. You get to drag the "neighbor" to all different parts of the circle and watch the angles change. Try it if you want a visual or extra practice!