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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

Or

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,

Angle of a circle

Angle PAQ is the exterior angle of the circle.

Examples

Example 1: Find the angle PAQ?

Angle of a circle

Solution:

Given:

Angle PSQ = 120°

Angle PRQ = 82°

According to the angle theorem,

Angle PAQ = ½ (Angle PSQ - Angle PRQ)

Angle PAQ = ½ (120 - 82)

Angle PAQ = ½ (38)

Angle PAQ = 19°

Thus, the exterior angle PAQ is equal to 19 degrees.

Example 2: Find the value of angle X?

Angle of a circle

Solution:

The circle has three intercepts with has three angle values.

220 + 45 + BD = 360

BD = 360 - 265

BD = 95

According to the angle theorem,

Angle X = ½ (Angle AED - Angle BD)

Angle X = ½ (220 - 95)

Angle X = ½ (125)

Angle X = 62.5°

Thus, the value of angle X is 62.5 degrees.

An interior angle is formed inside the circle, while the exterior angle is formed outside the circle.


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