Central Angle of a Circle Formula and Solved Examples

The formula for calculating the Central Angle of a Circle determines the angle formed between two radii of the circle. Alternatively, a central angle can be described as the angle created by the arc of a circle at the center of that circle. The radius vectors act as the arms of this central angle. To gain a clear grasp of the central angle of a circle formula, let's explore it through solved examples.

What is Central Angle of a Circle Formula?

In order to calculate the central angle using the central angle formula, you need to have two pieces of information: the measurement of the arc length that the central angle subtends at the center of the circle and the radius of the circle. The formula for the central angle of a circle is as follows:

Central angle, θ = (Arc length × 360º) / (2πr) degrees

Alternatively, it can be expressed as:

Central angle, θ = Arc length / r radians

Here, "r" represents the radius of the circle.

Also Check – Solid Shapes Formula

Examples on Central Angle of a Circle Formula

Example 1: Determine the central angle when the arc length measures 30 units, and the radius length is 15 units.

Solution:

To find the central angle, apply the central angle formula:

Central angle, θ = (Arc length × 360º) / (2πr) degrees

Substituting the values:

Central angle, θ = (30 × 360º) / (2π × 15) = 114.64º

Answer: The central angle for the given circle is θ = 114.64º.

Also Check – Sequence and Series Formula

Example 2: If the central angle of a circle is 90° and it forms an arc length of 12 cm, determine the radius of the circle.

Solution:

To find the radius of the circle, utilize the central angle formula:

Central angle, θ = (Arc length × 360º) / (2πr) degrees

Given central angle (θ) = 90° and arc length = 12 cm, we need to find the radius (r).

Rearrange the formula to solve for r:

r = (Arc length × 360º) / (2π × θ)

Substituting the values:

r = (12 cm × 360º) / (2π × 90º) = 7.64 cm

Answer: The radius of the given circle is 7.64 cm.