Angle of a circle

An angle is formed between the intersections of two lines. The angle of a circle or the central angle is the angle formed between the two radii, tangents, or chords. A radii is the radius of the circle. A tangent is a straight line touching or intersecting the circle at a specific point. It does not pass through the center of the circle. A chord is a line joining any two points on a circle.

Angle is represented by the symbol theta (θ).

Here, we will discuss the following:

Central angle
Interior angle
Exterior angle

Central angle

The central angle is the angle formed between the two radii in a circle. The vertex of the central angle lies at the center of the circle.

$Angle of a circle$

Where,

r is the radius of the circle

θ is the central angle

PQ is the length of the arc of the circle

Any line passing through the center of the circle is known as diameter of the circle.

$Angle of a circle$

Radius = Diameter/2

Diameter = 2Radius

Length of arc = 2πr × (θ/360)

θ = 360L/2πr

Where,

r is the radius of the circle

θ is the angle in degrees.

π is equal to 3.1415 or 22/7.

L is the length of the arc

Thus,

Central Angle θ = 360 x L/2πr

Central angle in radians is given by:

L = r θ

θ = L/r

Where,

r is the radius of the circle

θ is the angle in radians.

L = Arc length

Examples

Let's discuss two examples based on the central angle of a circle.

Example 1: Find the central angle of a circle with a radius 5m and length of the arc is 12m.

Solution:

Radius of the circle = 5m

Length of the arc = 12m

Central Angle θ = 360 x L/2πr

θ = 360 x 12/2π5

θ = 137.5

Thus, the central angle of the circle with a radius 5m and arc length of 2m is 137.5 degrees.

Example 2: Find the arc length of a circle with central angle and radius of 4 radians and 2 cm.

Solution:

θ = L/r

L = θr

L = 4 x 2

L = 8m

Thus, the length of the arc is 8 cm.

Example 3: Find the central angle of a circle with radius 4 cm and arc length 0.16m.

Solution:

Radius of the circle = 8 cm

Length of the arc = 0.16m

To find the central angle, we need to convert the different units of the radius and length to the same unit.

Length of the arc = 0.16 x 100 = 16cm

1m = 100 cm

Central Angle θ = 360 x L/2πr

θ = 360 x 16/2π8

θ = 114.6

Thus, the central angle of the circle with a radius 8cm and arc length of 0.16m is 114.6 degrees.

Example 4: Find the central angle in radians with a radius 6 m and arc length of 12 m.

Solution:

θ = L/r

θ = 12/6

θ = 2

Thus, the central angle of a circle is 2 radians.

Interior angle

The angle formed inside a circle is known as the interior angle of a circle. The vertex of the interior angle lies anywhere inside the circle or on the arc. Thus, the angle formed between the intersections of two lines inside a circle is the interior angle of a circle.

For example,

$Angle of a circle$

Angle AOB is the interior angle of the circle.

Examples

Example 1: Find the interior angle RPQ in a circle with radius 15cm and an angle of 80 degrees?

$Angle of a circle$

Solution:

OR = 15cm

According to the angle theorem, the angle RPQ is half the angle ROQ.

RPQ = ½ x 80

RPQ = 40 degrees

Thus, the value of angle is RPQ is 40 degrees.

Example 2: Find the interior angle AOB and angle COD in the below diagram.

$Angle of a circle$

Solution:

BOD is a straight line. The line OA intersects the line in two parts.

Angle AOD + Angle AOB = 180 °

128° + Angle AOB = 180 °

Angle AOB = 180 - 128

Angle AOB = 52°

Angle COD is the opposite angle of AOB. According to the angle theorem, the opposite angles are equal.

Thus, Angle COD = 52°

Exterior angle

The angle formed outside a circle is known as the interior angle of a circle. The vertex of the interior angle lies anywhere outside the circle or on the arc. Thus, the angle formed between the intersections of two lines outside a circle is the interior angle of a circle.

For example,