Central and Inscribed Angles: Definitions and Examples

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  • 0:07 Circles and Angles
  • 0:37 Central Angles
  • 1:44 Practice with Central Angles
  • 2:35 Inscribed Angles
  • 3:53 Practice with…
  • 5:50 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When we're working with circles, there are two key angles to know: central angles and inscribed angles. These angles have a few special theorems, which we'll discuss and practice using in this lesson.

Circles and Angles

Here's a clock. This particular time, 3 o'clock, is a memorable one. When I was in high school, it was what I always hoped to see when I looked up, waiting for the final bell to ring.

But this clock is more than just a way of telling time. It's also a circle. And the minute hand? It's a radius, or a line extending from the circle's center to its edge, or circumference. Yep, we're talking geometry here. If we take the hour hand and extend it to the circumference, then it's also a radius.


Radius of a clock
Radius on clock


Central Angles

These two lines show us three o'clock. And this angle here? It's called a central angle. A central angle is the angle formed by two radii in a circle.


Central Angle
Central Angle


How many total degrees are in a circle? In other words, in one hour, how many degrees does the minute hand travel? 360. So our central angle will be somewhere between 0, like this clock at 12 o'clock, and 360, like this clock also at 12 o'clock. If we put points on the circle like this, then we can name our angle. This is angle AOB.

Oh, and if we draw a line segment known as a chord right here, we make a triangle. A chord is just a line with endpoints on a circle. Now, what do we know about this triangle? First, if this chord is our second hand, then our clock is totally broken. But let's say it's just a chord.

The two radius lines are equal, since they're both traveling the same distance from the center of the circle. So this is an isosceles triangle. That means angle OAB and angle OBA are the same.


Isosceles Triangle
Isosceles Tri


Practice with Central Angles

Let's practice a bit with central angles. First, let's talk time. 3 o'clock is exactly one quarter of the circle, so this central angle is one quarter of 360, or 90 degrees. What about 6 o'clock? That's half the circle, so it's 180 degrees.

Now let's look at a triangle involving a central angle. What's the measure of the central angle in this circle? We can see that angle OAB is 50 degrees. Since both OA and OB are radius lines, they are equal, and this is an isosceles triangle. That means angle OBA is also 50 degrees. Since the sum of the interior angles of a triangle is 180, then the central angle, angle AOB, must be 180 - 100, or 80 degrees.

Inscribed Angles

Now here's a funny-looking clock. Did our minute and hour hands break? No, but this clock is completely non-functional as a timepiece. Instead of two radii, where the lines originate at the circle's center, we have two lines that start at the circumference.

This is an inscribed angle, or the angle formed by points on the circle's circumference. In this angle, which we call angle ACB, point C is the vertex and points A and B are the endpoints.


Inscribed Angle
Inscribed Angle


Inscribed angles have some special properties. First, with fixed endpoints, the measure of the inscribed angle remains the same regardless of the location of the vertex. So we can move C anywhere along the circumference and this angle doesn't change. That means that when we have two inscribed angles, as we see here, the angles are the same.


Two
Two Angles Inscribed


Second, when they share endpoints, the measure of an inscribed angle is half the measure of a central angle. So in this circle, angle AOB is twice angle ACB. This works as long as point C, the vertex of the inscribed angle, isn't on the arc formed by the central angle. If it is in that arc, well, it'll look kind of backwards. And then its measure would be the supplement of half the central angle.

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