What is the area of a sector with a central angle of 120 degrees?

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.

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.

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.

•  Central Angle Calculator 
•  Right Angle
•  Acute Angle 
•  Straight Line
•  Bisect

FAQs on Central Angle

What Is Central Angle?

As per central angle definition in geometry, it is the angle subtended by the arc of the circle at the center of the circle. The two radii make the arms of the angle. Central angle helps to know the proportion of the curve with respect to the circle.

How Do You Measure Central Angle of a Circle?

The central angle of a circle is measured in either degrees or radians. It is measured with the help of length of the arc and the length of the radius of the circle. The formula to measure central angle (in radians) = (Length of the arc)/(Length of the radius).

How Many Degrees is the Central Angle of a Circle?

The degree of a central angle is the angle made by the arc at the center of the circle.

What is Central Angle Theorem?

The central angle theorem states that the angle subtended by an arc length at the center of the circle is double the angle subtended at any point on the circumference of the circle.

What is The Central Angle of a Curve?

The central angle of a curve is the angle subtended by it at the center of the curve. The lines at the end of the curve connect at the center to form the central angle, and these lines are the arms of the angle or are the radii.

How Do You Find the Central Angle of an Arc?

To find the central angle of an arc, connect the ends of the arc with the center of the circle using the radius vectors. The angle between the two radii represents the central angle of the arc. Also, the formula to find the central angle of an arc is the length of the arc divided by the radius of the arc.

What is the Central Angle Made by a Semi-circle?

The central angle made by a semi-circle is 180°. The semi-circle represents half of a circle and hence the central angle of a semi-circle is half of the complete angle of a circle.

What is an Inscribed Angle?

The angle subtended by an arc at any point on the circle is called an inscribed angle. The arc of a circle is taken and any point on the circle distinct from the arc is taken. The ends of the arc is joined with the point on the circle, with a line. The angle between these lines and at the point on the circle is the inscribed angle.

What is the Difference Between Central Angle and Inscribed Angle?

Central angle is the angle subtended by an arc at the center of a circle. An inscribed angle is an angle subtended by an arc at any point on the circumference of a circle. The central angle is double the inscribed angle of the arc of a circle.

What is the Difference Between Reflex and Convex Angles?

Both reflex angle and convex angle can be central angles of a circle. A reflex angle is greater than 180 degrees and less than 360 degrees.  A convex angle is less than 180 degrees. For a given arc of a circle, the sum of its convex angle and reflex angle is equal to the complete angle. A complete angle is equal to 360 degrees. Convex angle + Reflex angle = Complete angle

What is the area of a sector with a central angle of 120?

How do you find the area of a sector with a central angle?

What is the length of the arc with a 120 central angle?