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In this lesson, you will learn about the definition and properties of a central angle. You will also discover what the Central Angle Theorem is and what the formula is for central angles. Test your new knowledge with a quiz.
Definition Of A Central Angle
A central angle is the angle that forms when two radii meet at the center of a circle. Remember that a vertex is the point where two lines meet to form an angle. A central angle’s vertex will always be the center point of a circle.
A central angle is formed at A.
Radii AB and AC meet at center A to form a central angle.
Reflex Versus Convex
It’s important to know that when two radii meet at the center of a circle to create a central angle, they also create another angle in the process. The convex central angle is the one that is shown in this diagram.
Convex Central Angle
A convex central angle is a central angle that measures less than 180°.
The reflex central angle that is formed is on the other side of the convex angle. Reflex angles measure more than 180° and less than 360°. Check out this second image to notice the reflex angle that is created on the other side of the convex central angle.
A reflex central angle is indicated and measures more than 180 degrees.
A reflex central angle is one that measures more than 180 and less than 360 degrees.
Central Angle Theorem
Before we understand what the central angle theorem is, we must understand what subtended and inscribed angles are, because they are a part of the definition. A subtended angle is an angle that is created by an object at a given outer position.
If you can’t wrap your head around that definition, picture this: You are standing on planet Earth and looking up at the sun. A triangle with three sides is formed. The three sides are the ray of the sun that goes from the top of the sun to your eyes, the ray that goes from the bottom of the sun to your eyes, and the height of the sun.
The sun is subtending an angle to your eyes.
The sun subtends an angle at your eyes.
We would say that the angle subtended by the sun to your eyes changes, although not by much, depending on if you move forward or backward. The angle will increase in size if you step forward and decrease if you step backward.
An inscribed angle is one that is subtended at a point on a circle by two identified points on a circle.
If you can’t wrap your head around that definition, picture this: Imagine that you are jogging around a perfect circular pond in your neighborhood. There are two neighbors, Ed and Tom, on the opposite side of the pond, close to each other, but at different points. Now picture a line drawn from Ed to you and one from Tom to you. The angle that is created when both lines are drawn to you is an inscribed angle, because all three points are directly on the circle.
The angle at your location is the inscribed angle.
The lines from Ed and Tom to you are inscribing an angle at you.
Now that you understand what subtended and inscribed angles means, we can move forward to the Central Angle Theorem.
The Central Angle Theorem states that the central angle subtended by two points on a circle is always going to be twice the inscribed angle subtended by those points.
In the three diagrams you can see now, you can see the inscribed angle ADB is always half the measurement of the central angle ACB, no matter where the vertex of the angle (point D) is on the circle.
Central Angle Theorem Diagram 1
Central Angle Theorem diagram 2
Central Angle Theorem diagram 3
The Central Angle Theorem is always true unless the vertex of the inscribed angle (point D) lies on the minor arc instead of the major arc.
Let’s look at the major arc vs. minor arc first:
The arc AB surrounding the shaded blue area is the Major Arc.
Major Arc
The arc AB surrounding the shaded blue area is the Minor Arc.
Minor Arc
The length between points A and B is called the arc length (we will need this information later).
Now that we know the difference between the major and minor arcs, we can use the following formula to determine the measurement of the inscribed angle when the vertex of the inscribed angle (point D) lies on the minor arc instead of the major arc.
Inscribed angle is 180 minus the central angle divided by 2
Central Angle Formula
Depending on if you want to find the measurement of the central angle or the arc length, the following forms of the central angle formula will be necessary:
To find the central angle measurement, when you know the radius and arc length, use this formula:
Central Angle equals the Arc Length times 360 divided by 2 times Pi times the Radius
 
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