how to find the central angle

Central Angle

Central Angle is the angle formed by two arms with the center of a circle as the vertex. The two arms form two radii of the circle intersecting the arc of the circle at different points. Central angle helps 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.

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

The angle subtended by an arc at the center of a circle is the central angle. The radius vectors form the arms of the angle. In other words, an angle whose vertex is the center of a circle with the two radii lines as its arms that intersect at two different points. 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.

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.

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.

Angle Subtended by an Arc
∠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 the two radii) and the radius length. The steps given below shows how to calculate central angle in radians.

There are three simple steps to find 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.

Construction of central angle - step 1

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.

Constructing Central Angle Step 2

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

Important topics

Example 1: Sam measures the angle in a triangle with the help of a protractor as 60º. Convert the angle into radian measure.
Solution:

The given angle of 60° is in sexagesimal measure.

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

Radian = π/3

Therefore, the angle is π/3 radians.
Example 2: Larry drew a circle and cut it into four equal parts using two diameters. How can you help Larry to measure the central angle or inscribed angle of each part of the circle?
Solution:

Larry cuts the circle into four equal parts.

Complete angle in a circle = 360°

Angle of each quadrant = 360°/4

= 90°
Therefore, the central angle of a quadrant is 90°.
Example 3: Sally marks an arc of length 8 inches and measures its central angle as 120 degrees. What is the radius of the arc?
Solution:

Radius of the arc = 8 inches

Central Angle = 120°

Central angle = (length of arc × 360°)/(2 π × radius)

radius = (length of 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 uses a compass to draw an arc of length 11 inches and a radius of 7 inches. Without using a protractor, how can Jim calculate the angle of this arc?
Solution:

Length of 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 angle of the arc is 90°.
Example 5: George wants to create a garden in the shape of a sector of radius 42 feet and having a central angle of 120 degrees. Calculate the area of the grass which is required to cover the garden.

Solution:

Given that the shape of the garden is a sector.

Radius = 42 feet

Central Angle = 120°

The area of the grass required to cover the garden is the same as the area of the sector.

Area of 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 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.

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 Central Angle Definition in Geometry?

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.

How Many Degrees is the Central Angle of a Circle?

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

What is Central Angle Theorem?

The central angle theorem states that the angle subtended by an arc at the center of the circle is double the angle subtended at any point on the circumference of 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 length of the radius of the circle. The formula to measure central angle (in radians) = (Length of the arc)/(Length of the radius).

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

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

What is an Inscribed Angle?

The angle subtended by an arc at any point on the circle is called an 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. Inscribed angle is an angle subtended by an arc at any point on the circumference of a circle.