Central angle is the angle created by two arms v the facility of a circle together the vertex. The 2 arms type two radii the the one intersecting the arc of the circle at various points. Main angle helps to divide a circle into sectors. A slice of pizza is a great example of main angle. A pie chart is consisted of of a variety of sectors and also helps to represent different quantities. A protractor is a basic example that a sector with a main angle of 180º. Central angle can also be identified as the angle created by an arc that the circle in ~ the center of the circle.

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1.Definition of main Angle
2. Central edge Theorem
3.How come Find main Angle?
4.Solved instances on main Angles
5.Practice inquiries on main Angles
6.FAQs on central Angles

Definition of central Angle

The edge subtended by an arc in ~ the facility of a circle is the main angle. The radius vectors type the arms of the angle. In other words, one angle who vertex is the center of a one with the 2 radii lines together its arms that crossing at two various points. As soon as these 2 points room joined they form an arc. Central angle is the edge subtended by this arc in ~ the facility of the circle.



Here O is the center of the circle, ab is the arc and, OA is a radius and OB is one more radius that the circle.

Central Angle= (fracs imes 360^02 pi r)Here "s" is the size of the arc and "r" is the radius of the circle. This is the formula for finding main angle in degrees. For finding the central angle in radians, we need to divide the arc size by the length of the radius the the circle.

Central angle Theorem

Theorem: The edge subtended by an arc at the center of the one is dual the angle subtended by it at any other suggest on the circumference of the circle.


The central angle theorem claims that the central angle of a one is dual the measure of the edge subtended through the arc in the various other segment of the circle.

∠AOB = 2 × ∠ACB 

Central angle = 2 × angle in various other segment

How to Find central Angle?

The main angle is the edge between any type of two radii that a circle. To discover the main angle we require to find the arc length (which is the distance in between the two points the intersection the the the 2 radii) and the radius length. The actions given below shows exactly how to calculate central angle in radians.

There space three simple steps to find the main angle.

Identify the end of the arc and also the center of the circle (curve). AB is the arc the the circle and also O is the center of the circle.


Join the ends of the arc v the center of the circle. Also, measure the length of the arc and the radius. Here abdominal is the length of the arc and also OA and OB room the radii of the circle.


Divide the length of the curve v the radius, to get the main angle. By using the formula presented below, us will uncover the value of the central angle in radians. 

( extCentral Angle = dfrac extLength of the ArcRadius)

Important notes

The central angle that a circle is measured in radian measure and also sexagesimal measure.

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

Radian × (180/π) = Sexagesimal

Topics related to central Angle

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

Important topics

Solved Examples

Example 1: Sam procedures the edge in a triangle with the aid of a protractor as 60º. Convert the angle right into radian measure.Solution:

The offered angle of 60° is in sexagesimal measure.

Radian = π/180° × Sexagesimal Radian = π/180° × 60° 

Radian = π/3Therefore, the angle is π/3 radians.

Example 2: Larry attracted a circle and also cut it into 4 equal parts using two diameters. How have the right to you help Larry to measure up the central angle or inscriptions angle of each component of the circle?Solution: 

Larry cut the one into 4 equal parts.

Complete edge in a one = 360°

Angle of every quadrant = 360°/4

= 90°Therefore, the main angle that a quadrant is 90°.

Example 3: Sally point out an arc of length 8 inches and also measures its main angle together 120 degrees. What is the radius that the arc?Solution: 

Radius of the arc = 8 inches

Central edge = 120°

Central edge = (length the arc × 360°)/(2 π × radius)

radius = (length the arc × 360°)/(2 π × central angle) 

radius =(8 × 360°) / (2 π × 120°)

radius = 12/π

Therefore, the radius is 12/π inches.

Example 4: Jim offers a compass to attract an arc of length 11 inches and also a radius the 7 inches. Without using a protractor, how can Jim calculation the edge of this arc?Solution: 

Length that the arc = 11 inches

Radius of the arc = 7 inches

 Angle of the arc = (length of arc × 360°)/(2 π r)

Angle = (11 × 360°)/ (2 × 22/7 × 7)

Angle = 90°Therefore, the edge of the arc is 90°.

Example 5: George desires to develop a garden in the shape of a ar of radius 42 feet and having actually a main angle the 120 degrees. Calculation the area of the grass i m sorry is compelled to covering the garden. Solution: 

Given the the shape of the garden is a sector.

Radius = 42 feet

Central edge = 120°

The area that the grass required to sheathe the garden is the exact same as the area that the sector.

Area that the sector = θ/360° × π r2

Area =(120°/360°) × π × 422

Area = 1/3 × 22/7 × 42 × 42 

 Area = 22 × 2 × 42 

 Area = 1848 Therefore, the area of the sector is 1848 square feet.

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FAQs on central Angle

How carry out You uncover the main Angle of one Arc?

To find the central angle of one arc, attach the end of the arc through the center of the circle using the radius vectors. The angle in between the 2 radii to represent the central angle the the arc.

What is the Difference in between Reflex and Convex Angles?

Both reflex angle and convex angle deserve to be central angles the a circle. A reflex edge is higher than 180 degrees and also less than 360 degrees. A convex edge is much less than 180 degrees.For a given arc that a circle, the sum of its convex angle and also reflex angle is equal to the finish angle. A complete angle is equal to 360 degrees. Convex edge + Reflex edge = finish angle

What Is central Angle an interpretation in Geometry?

As per main angle meaning in geometry, the is the angle subtended by the arc that the circle in ~ the facility of the circle. The 2 radii do the eight of the angle.

How plenty of Degrees is the central Angle that a Circle?

The degrees of a main angle is the edge made by the arc at the center of the circle.

What is The main Angle of a Curve?

The main angle that a curve is the angle subtended by it at the center of the curve.

What is main Angle Theorem?

The main angle theorem says that the edge subtended by one arc at the center of the one is twin the angle subtended in ~ any point on the circumference of the circle.

How do You Measure main Angle the a Circle?

The central angle that a one is measured in either degrees or radians. That is measured v the aid of length of the arc and also length of the radius the the circle. The formula to measure central angle (in radians) = (Length the the arc)/(Length of the radius).

What is the main Angle make by a Semi-circle?

The central angle do by a semi-circle is 180°.

What is an inscriptions Angle?

The angle subtended by an arc at any allude on the one is dubbed an enrolled angle.

See more: How Much Is 6 Ounces In Cups ? Convert 6 Ounces To Cups

What is the difference Between central Angle and Inscribed Angle?

Central edge is the angle subtended by an arc at the center of a circle. Inscribed angle is an edge subtended by an arc in ~ any suggest on the circumference of a circle.