write the formula for the area of a segment in a circle, a circle of radius r, given that the - Maths - Areas Related to Circles

Question

write the formula for the area of a segment in a circle, a
circle of radius r, given that the sector angle is theta (in
degrees) the answer to this is [pi theta/360- sin theta /2 cos
theta /2] r2 as given in r d sharma I can't understand how the
formula derived
experts plzzz help me in dis question

Manmeet Singh · Accepted Answer

Given: A circle with center O of radius r âˆ AOC = Î¸ and âˆ AOD = $\frac{θ}{2}$

To find: Area of segment ABC Solution: Area of segment ABC = area of sector OABC â€“ area of Î”OPQ The area of sector OABC = $\frac{θ}{360^{°}} \times π r^{2}$ In Î”OAD, $\cos \frac{θ}{2} = \frac{O D}{O A} \Rightarrow O D = O A \times \cos \frac{θ}{2} = r \cos \frac{θ}{2} A l s o, \sin \frac{θ}{2} = \frac{A D}{O A} \Rightarrow A D = O A \times \sin \frac{θ}{2} = r \sin \frac{θ}{2} A C = 2 A D = 2 r \cos \frac{θ}{2} S i n c e l i n e p e r p e n d i c u l a r f r o m c e n t e r t o c h o r d b i \sec t s t h e c h o r d A r e a o f t r i a n g l e O A C = \frac{1}{2} \times A C \times O D = \frac{1}{2} \times 2 r \cos \frac{θ}{2} \times r \cos \frac{θ}{2} = r^{2} \sin \frac{θ}{2} \cos \frac{θ}{2} H e n c e, a r e a o f s e g m e n t = \frac{θ}{360^{°}} \times π r^{2} - r^{2} \sin \frac{θ}{2} \cos \frac{θ}{2}$ Hence proved