Central Angle In Geometry - Definition, Formulae, Examples

Central Angle

Central Angle is the angle formed by two arms and having the vertex at the center of a circle. The two arms form two radii of the circle intersecting the arc of the circle at different points. Central angle is helpful to divide a circle into sectors. A slice of pizza is a good example of central angle. A pie chart is made up of a number of sectors and helps to represent different quantities.

A protractor is a simple example of a sector with a central angle of 180º. Central angle can also be defined as the angle formed by an arc of the circle at the center of the circle. Let us learn more about the central angle theorem, and how to find central angle, with the help of examples, FAQs.

1.	Definition of Central Angle
2.	Central Angle Theorem
3.	How To Find Central Angle?
4.	Solved Examples on Central Angles
5.	Practice Questions on Central Angles
6.	FAQs on Central Angles

Definition of Central Angle

Central angle is the angle subtended by an arc of a circle at the center of a circle. The radius vectors form the arms of the central angle. In other words, it is an angle whose vertex is the center of a circle with the two radii lines as its arms, that intersect at two different points on the circle. When these two points are joined they form an arc. 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 and, OA is a radius and OB is another radius of the circle. The central angle of a circle formula is as follows.

Central Angle= \(\frac{s \times 360^0}{2 \pi r}\) Here "s" is the length of the arc and "r" is the radius of the circle. This is the formula for finding central angle in degrees. For finding the central angle in radians, we have to divide the arc length by the length of the radius of the circle.

Central Angle Theorem

Theorem: The angle subtended by an arc at the center of the circle is double the angle subtended by it at any other point on the circumference of the circle.

OR

The central angle theorem states that the central angle of a circle is double the measure of the angle subtended by the arc in the other segment of the circle.

       ∠AOB = 2 × ∠ACB 

Central Angle = 2  ×  Angle in other segment

How To Find Central Angle?

The central angle is the angle between any two radii of a circle. To find the central angle we need to find the arc length (which is the distance between the two points of intersection of the two radii with the circumference) and the radius length. The steps given below shows how to calculate 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.

• Join the ends of the arc with the center of the circle.  Also, measure the length of the arc and the radius. Here AB is the length of the arc and OA and OB are the radii of the circle.

            

• Divide the length of the curve with the radius, to get the central angle. By using the formula shown below, we will find the value of the central angle in radians. 

        \(\text{Central Angle} = \dfrac{\text{Length of the Arc}}{Radius}\)

Important Notes

• The central angle of a circle is measured in radian measure and sexagesimal measure.

• The unit of radian measure is radians and the unit of sexagesimal measure is degrees.

• Radian × (180/π) = Sexagesimal

Topics Related to Central Angle

Check out these interesting articles to know about central angle and its related topics.

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