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.

## 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.

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.

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:

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:

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:

## Key Terms

**Central angle** – the angle that forms when two radii meet at the center of a circle**Convex 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 position**Inscribed angle** – an angle that is subtended at a point on a circle by two identified points on a circle**Central 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