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 angle subtended by an arc 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 intercepted arc on the circle's circumference.
2. Central angles are used to calculate the area of a circle sector, which is the region bounded by two radii and the intercepted arc.
3. The sum of the central angles of a circle is always $360^\circ$.
4. Central angles are important in solving problems involving the relationships between circles, arcs, and sectors.
5. The length of the intercepted arc is proportional to the measure of the central angle.

Review Questions

• Explain how the measure of a central angle is related to the measure of the intercepted arc on a circle's circumference.
  • The measure of a central angle is equal to the measure of the intercepted arc on the circle's circumference. This relationship is fundamental in solving geometry problems involving circles, as the central angle and the arc length are directly proportional. For example, if a central angle has a measure of $45^\circ$, the intercepted arc will also have a measure of $45^\circ$ of the circle's total circumference.
• Describe how central angles are used to calculate the area of a circle sector.
  • The area of a circle sector, which is the region bounded by two radii and the intercepted arc, is directly related to the measure of the central angle. The formula for the area of a circle sector is $\frac{1}{2}r^2\theta$, where $r$ is the radius of the circle and $\theta$ is the measure of the central angle in radians. This formula allows us to determine the area of a circle sector given the radius of the circle and the measure of the central angle.
• Analyze the relationship between the sum of all central angles in a circle and the total measure of the circle.
  • The sum of all central angles in a circle is always $360^\circ$. This is because the central angles, by definition, divide the circle into equal parts, and the total measure of the circle is $360^\circ$. This relationship is crucial in understanding the properties of circles and solving problems involving the relationships between central angles, arcs, and sectors. It also highlights the importance of central angles in the study of circle geometry.

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.

Subtended: To be contained or spanned by an angle or arc.