In what subtended and inscribed angles means, we

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.

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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.

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.</p>
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<h2>Central Angle Theorem</h2>
<p>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 <b>subtended angle</b> 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.</p>
<p> 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.</p>
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The sun is subtending an angle to your eyes.
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.
Central Angle equals the Arc Length times 360 divided by 2 times Pi times the Radius

So, the central angle is essentially the arc length multiplied by 360, the degrees of a full circle, divided by the circumference of the circle.To find the arc length, when you know the central angle measurement and radius, use this formula:

Arc Length Formula

As you can see, the arc length is simply the circumference of a circle (2;R) multiplied by the ratio of the arc angle to the full 360 angle of a circle.

Lesson Summary

To sum up, a central angle is the angle that forms when to radii meet at the center of a circle. There are two types of central angles. A convex central angle, which is a central angle that measures less than 180 degrees and a reflex central angle, which is a central angle that measures more than 180 degrees and less than 360 degrees. These are both part of a complete circle. One can’t exist without the other, meaning that if the convex central angle measures at 60 degrees, the reflex central angle would be 300.

They add up to a complete 360.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. But in order to fully understand that, you need to understand the difference between a subtended angle, which is an angle that is created by an object at a given outer position, and an inscribed angle, which is an angle that is subtended at a point on a circle by two identified points on a circle. The formula for the central angle, essentially, is the arc length multiplied by 360, the degrees of a full circle, divided by the circumference of the circle.

It can be looked at this way:

Central Angle equals the Arc Length times 360 divided by 2 times Pi times the Radius

Key Terms

Central Angle

Central angle – the angle that forms when two radii meet at the center of a circleConvex central angle – a central angle that measures less than 180°Reflex central angle – a central angle that measures more than 180° and less than 360°Subtended angle – an angle that is created by an object at a given outer positionInscribed angle – an angle that is subtended at a point on a circle by two identified points on a circleCentral Angle Theorem – theorem which 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

Learning Outcomes

Use this lesson to increase your understanding of central angles as you prepare to:

  • Give the definition of central angle
  • Differentiate between convex and reflex central angles
  • Use the central angle theorem to find the degree of an angle subtended by two points given the inscribed angle by those points
  • Apply the central angle formula to find a central angle given the radius and arc length of the circle
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