Central Angle of a Circle Formula

The central angle of a circle formula calculates the angle between two radii of a circle. A central angle can also be defined as an angle subtended by the arc of a circle at the center of the circle. The radius vectors form the arms of the central angle. Let us understand the central angle of a circle formula using solved examples.

What is Central Angle of a Circle Formula?

To calculate the central angle using the central angle formula, we require the measure of the arc length that subtends the central angle at the center and the radius of the circle. The central angle of a circle formula is given as,

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

or,

Central angle, θ = Arc length/r radians

where r is the radius of the circle.

Let us see how to use the central angle formula in the following solved examples section.

Examples on Central Angle of a Circle Formula

Example 1: Find the central angle, where the arc length measurement is about 30 units and the length of the radius is 15 units.

Solution:

To find: Central angle

Using the central angle of a circle formula,

\( \text{Central angle}, \theta = \dfrac{\text{Arc length} \times 360^\circ}{2 \pi \text r} \text{degrees}\)

\( \text{Central angle}, \theta = \dfrac{30 \times 360^\circ}{2 \pi \times 15} \) = 114.64º

Answer: Central angle for the given circle, θ = 114.64º.

Example 2: If the central angle of a circle is 90° and the arc length formed is 12 cm then find out the radius of the circle.

Solution:

To find: The radius of the given circle

Using the central angle formula,

\( \text{Central angle}, \theta = \dfrac{\text{Arc length} \times 360^\circ}{2 \pi \text r} \text{degrees}\)

\( 90^\circ = \dfrac{ 12 \times 360^\circ}{2 \pi \text r} \)

\( r = \dfrac{ 12 \times 360^\circ}{2 \pi \times 90^\circ} = 7.64 \text{cm} \)

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