How to Find the Central Angle of a Circle?

A central angle is an angle formed by two arms whose vertex is at the center of the circle. In this step-by-step guide, you learn more about finding the central angle.

How to Find the Central Angle of a Circle?

The central angle is useful for dividing a circle into sections. A slice of pizza is a good example of a central angle. A pie chart consists of several sections and helps to display different values.

Step by step guide to finding the central angle of a circle

The central angle is the angle subtended by an arc of a circle at the center of a circle. Radius vectors form the arms of the central angle. In other words, it is an angle whose vertex is the center of a circle whose arms are two radii lines that intersect at two different points in the circle. When these two points are connected, they form an arc. The central angle is the angle subtended by this arc at the center of the circle.

Here \(O\) is the center of the circle, \(AB\) is the arc, \(OA\) is a radius, and \(OB\) is another radius of the circle.

How to find the central angle?

The central angle is the angle between the two radii of the circle. To find the central angle, we need to find the length of the arc, and the length of the radius. The following steps show how to calculate the central angle in radians.

There are three simple steps to finding the central angle:

  • Identify the ends of the arc and the center of the circle (curve). \(AB\) is the arc of the circle and \(O\) is the center of the circle.
  • Connect the end of the arc with the center of the circle. Also, measure the length of the arc and its radius. Here \(AB\) is the arc length and \(OA\) and \(OB\) are the radii of the circle.
  • Divide the length of the curve by the radius to get the central angle. Using the following formula, we will find the value of the central angle in radians.

\(\color{blue}{Central\:Angle=\frac{Length\:of\:the\:Arc}{Radius}}\)

Finding the Central Angle of a Circle – Example 1:

If the arc length is \(8\) inches and the central angle is \(120\) degrees, find the radius of the arc.

Solution: 

\(Radius\:of\:the\:arc= 8\space inches\)

\(Central\:angle= 120°\)

\(Central\:angle\:=\frac{\left(length\:of\:arc\:×\:360°\right)}{\left(2\pi \:×\:radius\right)}\)

\(radius\:=\frac{\left(length\:of\:arc\:×360°\right)}{\left(2\pi \:×\:Central\:angle\right)}\)

\(radius\:=\frac{\left(8\:×\:360°\right)}{\left(2\pi \:×120°\right)}\)

\(radius\:=\frac{12}{\pi \:}\)

Exercises for Finding the Central Angle of a Circle

  • From the diagram shown, find the \(m∠BOD\) measure. 
  • From the diagram shown, find the \(m∠arc MNK\) measure. 
  • If the arc length measurement is about \(21\space cm\) and the length of the radius measures \(10\space cm\), find the central angle.
This image has an empty alt attribute; its file name is answers.png
  • \(\color{blue}{60°}\)
  • \(\color{blue}{234°}\)
  • \(\color{blue}{120.38°}\)

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