Central Angle

Definition

A central angle is an angle formed by two radii of a circle that intersect at the center of the circle. It represents the measure of the arc that the angle subtends on the circle's circumference.

5 Must Know Facts For Your Next Test

1. The measure of a central angle is equal to the measure of the arc it subtends on the circle's circumference.
2. Central angles are measured in degrees, with a full circle measuring 360 degrees.
3. The sum of the measures of all central angles in a circle is always 360 degrees.
4. Central angles can be used to find the measure of an arc or the measure of a sector of a circle.
5. The formula for the measure of a central angle is: $\theta = \frac{s}{r}$, where $\theta$ is the measure of the central angle, $s$ is the length of the arc, and $r$ is the radius of the circle.

Review Questions

• Explain how the measure of a central angle is related to the measure of the arc it subtends on the circle's circumference.
  • The measure of a central angle is directly proportional to the measure of the arc it subtends on the circle's circumference. Specifically, the measure of the central angle, in degrees, is equal to the measure of the arc, also in degrees. This relationship allows us to use central angles to find the measure of arcs and vice versa, which is particularly useful in solving geometry problems involving circles.
• Describe how the sum of the measures of all central angles in a circle is always 360 degrees.
  • The sum of the measures of all central angles in a circle is always 360 degrees because a full circle represents a complete revolution, or 360 degrees. This is because central angles are formed by two radii that intersect at the center of the circle, and the total number of degrees in a full revolution around the circle's center is 360 degrees. This property of central angles is important for understanding the relationships between different parts of a circle and solving problems involving circular geometry.
• Analyze the formula for the measure of a central angle, $\theta = \frac{s}{r}$, and explain how it can be used to find the measure of an arc or a sector of a circle.
  • The formula for the measure of a central angle, $\theta = \frac{s}{r}$, where $\theta$ is the measure of the central angle, $s$ is the length of the arc, and $r$ is the radius of the circle, can be used to solve for any of these three variables if the other two are known. For example, if you know the measure of the central angle and the radius of the circle, you can solve for the length of the arc. Alternatively, if you know the length of the arc and the radius, you can solve for the measure of the central angle. This formula is a powerful tool for analyzing and solving problems involving the relationships between central angles, arcs, and the dimensions of a circle.

Related terms

Radius: A line segment that connects the center of a circle to a point on the circle's circumference.

Arc: A portion of the circumference of a circle bounded by two points on the circle.

Circumference: The distance around the outside of a circle, calculated as $2\pi r$, where $r$ is the radius of the circle.