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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.


$\theta$ = 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
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