Area of a Segment of a Circle Formula

A circular segment is a region of a circle which is created by breaking apart from the rest of the circle through a secant or a chord. In other words it is two equal halves that are divided by the circle’s arc and connected through chord by the endpoints of the arc. Formulas to calculate the area of a segment of a circle is given below.  According to the definition, the part of circular region which is enclosed between a chord and corresponding arc is known as segment of a circle. The segment portraying a larger area is known as the major segment and the segment having smaller area is known as a minor segment.

Area of a segment of a circle

Formula of area of a segment

\(Area_{radians}=\frac{1}{2}\;r^{2}(\Theta -sin\Theta )\)

\(Area_{degrees}=\frac{1}{2}\;r^{2}\left (\frac{\Pi }{180}\Theta – sin \Theta \right )\)

 

Solved Examples

Question 1:

Find the area of a segment of a circle with a central angle of 75 degrees and a radius of 5 inches.

Solution:

Given,
$\theta$ = 75
radius = r = 5 inches
$Area_{radians}$ = $\frac{1}{2}$$r^{2}$($\theta$ – sin$\theta$)
= $\frac{1}{2}$ $\times$ $5^{2}$ $\times$ (75 – sin75)
= $\frac{1}{2}$ $\times$ 25 $\times$ {75 – (-0.3877)}
= $\frac{1}{2}$ $\times$ 25 $\times$ (75 + 0.3877)
= $\frac{1}{2}$ $\times$ 25 $\times$ (75.3877)
= $\frac{1}{2}$ $\times$ 1884.6925
= 942.34

Practise This Question

A spring is loaded with two blocks m1 and m2 where m1 is rigidly fixed with the spring and m2 is just kept on the block m1 as shown in the figure. The maximum energy of oscillation that is possible for the system having the block m2 in contact with m1 is