Understanding Central Angles | Definition, Formula, and Applications in Math

Central Angle

A central angle is an angle whose vertex is at the center of a circle

A central angle is an angle whose vertex is at the center of a circle. It is formed by two radii (plural of radius) of the circle, with one radius extending from the center to a point on the circle’s circumference and the other radius extending to a different point on the circumference. The measure of a central angle is equal to the measure of the arc intercepted by the angle, and it is always measured in degrees or radians.

To find the measure of a central angle, we can use the following formula:

Measure of Central Angle = (Arc Length / Circle’s Circumference) * 360°

In this formula, the arc length is the length of the portion of the circle’s circumference intercepted by the central angle, and the circle’s circumference is the total distance around the circle.

For example, let’s say we have a circle with a circumference of 20 units, and we want to find the measure of a central angle that intercepts an arc of length 5 units. Using the formula, we can calculate:

Measure of Central Angle = (5 / 20) * 360°
Measure of Central Angle = (1/4) * 360°
Measure of Central Angle = 90°

So, the measure of the central angle in this case is 90°.

Central angles are important in various mathematical concepts, including geometry, trigonometry, and calculus, as they play a significant role in determining relationships and properties of circles and circular arcs. They are often used in problems involving sectors, segments, and other geometric measurements within circles.

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