Central Angle of a Circle Formula

A circle is a round shape figure whose boundary is equidistant from its center point. The distance between the center point and the boundary is known as the radius of the circle. The angle formed by the two radii of the circle is known as the central angle. The circle completes an angle of 360°. The value of the central angle of a circle lies between 0 and 360°. In radians, the value of the central angle of a circle lies between 0 and 2π.  

Central Angle of a Circle Formula

If the two points on the circle are exactly opposite to each other, it forms an internal angle of 180°. Otherwise, the angle formed is always less than 180° or π radians. A semicircle subtends the maximum angle of 180°. Similarly, a quad circle subtends the maximum angle of 90°.  

The angle less than 180° is called convex angle and the angle greater than 180° is called a reflex angle. An example of central angle theta is shown below:

O is the center point of the circle and AB is the arc.  

The formula to calculate the center angle of a circle is given by:

Length of arc = 2πr × (θ/360)

θ = 360L/2πr

Where,

r is the radius of the circle

Theta is the angle in degrees.  

L = Arc length  

Or

L = r θ

θ = L/r

Where,

r is the radius of the circle

Theta is the angle in radians.  

L = Arc length  

Sample Problems

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

Solution:  

The formula to calculate the central angle in radians is given by:

θ = L/r

Where,

L is the arc length

r is the radius

L = 8m

r = 5m

θ = 8/5 =  1.6 radians

Thus, the central angle of the circle of radius 5m and arc length of 8m is 1.6 radians.

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

Solution:  

The formula to calculate the central angle in degrees is given by:

θ = 360L/2πr

Where,

L is the arc length

r is the radius

L = 4m

r = 2m

θ = 360 × 4/ (2 × 3.1415 × 2)

(π = 3.1415)

θ = 114.6°

Thus, the central angle of the circle of radius 2m and arc length of 4m is 114.6°.

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

Solution:  

The formula to calculate the central angle in radians is given by:

θ = L/r

where,

L is the arc length

r is the radius

L = 18m

r = 6m

θ = 18/6 = 3 radians

Thus, the central angle of the circle of radius 6m and arc length of 18m is 3 radians.

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

Solution:  

The formula to calculate the central angle in degrees is given by:

θ = 360L/2πr

where,

L is the arc length

r is the radius

L = 30cm

r = 10cm

θ = 360 × 30/ (2 × 3.1415 × 10)

(π = 3.1415)

θ = 171.9°

Thus, the central angle of the circle of radius 10cm and arc length of 30cm is 171.9°.

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

Solution:  

The formula to calculate the central angle in radians is given by:

θ = L/r

Where,

L is the arc length

r is the radius

L = 280cm

r = 7m

Both the dimensions are in different units. We need to first convert the dimensions into the same unit.

L = 2.8m 

(1m = 100cm)

r = 7m

θ = 2.8/7 =  0.4 radians

Or

L = 280cm

r = 7m = 700cm

(1m = 100cm)

θ = 280/700 =  0.4 radians

Thus, the central angle of the circle of radius 7m and arc length of 280cm is 0.4 radians.