Central Angle of Circle Formula

Last Updated : 25 Mar, 2026

A central angle is an angle formed at the center of a circle by two radii, with its vertex at the center. It represents the angle subtended by an arc at the center and determines the portion of the circle covered by that arc.

1

Formula

  • In Degrees:
    Central Angle = (s × 360°) / (2πr)
  • In Radians:
    Central Angle = s / r

where, s = arc length and r = radius of the circle.

Central Angle Theorem

The Central Angle Theorem states that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference of the circle.

2-

For a circle with center O and points A, B, and C on the circumference, the angle formed at the center is always double the angle formed at the circumference by the same arc AB.

∠AOB = 2 × ∠ACB

Finding the Central Angle

The central angle is the angle formed between two radii of a circle at its center. It depends on the length of the arc and the radius of the circle. By using these two values, we can easily determine the central angle in radians.

Follow the simple steps below to calculate the central angle:

Step 1: Identify the arc (AB) and the center (O) of the circle.

3-

Step 2: Join points A and B to the center O to form two radii (OA and OB).

4-

Step 3: Measure the arc length (s) and the radius (r).

Step 4: Apply the formula: Central Angle = s / r (where s is arc length and r is radius).

Solved Examples

Question 1: Find the central angle in radians of the circle of radius 5 m and arc length of 8 m.

Solution:  

The formula to calculate the central angle in radians is given by:
θ = L/r
Where, L is the arc length and r is the radius.
L = 8 m, r = 5 m
θ = 8/5 = 1.6 radians
Thus, the central angle of the circle is 1.6 radians.

Question 2: Find the central angle in degrees of the circle of radius 2 m and arc length of 4 m.

Solution:  

The formula to calculate the central angle in degrees is given by:
θ = 360L / 2πr
Where, L is the arc length and r is the radius.
L = 4 m, r = 2 m
θ = 360 × 4 / (2 × 3.1415 × 2) (π = 3.1415)
θ = 114.6°
Thus, the central angle of the circle is 114.6°.

Question 3: Find the central angle in radians of the circle of radius 6 m and arc length of 18 m.

Solution:  

The formula to calculate the central angle in radians is given by:
θ = L/r
where, L is the arc length and r is the radius.
L = 18 m, r = 6 m
θ = 18/6 = 3 radians
Thus, the central angle of the circle is 3 radians.

Question 4: Find the central angle in degrees of the circle of radius 10 cm and arc length of 30 cm.

Solution:  

The formula to calculate the central angle in degrees is given by:
θ = 360L / 2πr
where, L is the arc length and r is the radius.
L = 30 cm, r = 10 cm
θ = 360 × 30 / (2 × 3.1415 × 10) (π = 3.1415)
θ = 171.9°
Thus, the central angle of the circle is 171.9°.

Question 5: Find the central angle in radians of the circle of radius 7 m and arc length of 280 cm.

Solution:  

The formula to calculate the central angle in radians is given by:
θ = L/r
Where, L is the arc length and r is the radius.
L = 280 cm, r = 7 m
Both the dimensions are in different units, so convert them into the same unit.

L = 2.8 m (1 m = 100 cm), r = 7 m
θ = 2.8/7 = 0.4 radians

Comment

Explore