## Central angle

### An angle that intersects a circle in two points and has its vertex at the center of the circle

Central angle is an angle whose vertex is located at the center of a circle and whose sides intersect the circle at two distinct points. The measure of a central angle is equal to the measure of the arc that it intercepts on the circle. For example, if a central angle intercepts an arc of 60 degrees on a circle, then the central angle itself has a measure of 60 degrees.

Central angles are important in geometry because they are used to define and calculate other important concepts, such as the inscribed angle and the arc length. In particular, the measure of an inscribed angle is half the measure of its corresponding central angle. Additionally, the arc length intercepted by an inscribed angle is proportional to the measure of the central angle.

Central angles can also be used to calculate the area of a sector of a circle. The area of a sector is equal to the fraction of the circle’s total area that is intercepted by the central angle. Specifically, if the central angle has a measure of theta degrees and the circle has a radius of r, then the area of the sector is (theta/360) times pi times r squared.

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