Question

Write the formula for the area of a segment in a circle of radius r given that the sector angle is \[\theta\] (in degrees). 

Answer 1

In this figure, centre of the circle is O, radius OA = r and ∠ AOB=θ

We are going to find the area of the segment AXB.

Area of the segment AXB=Area of the sector `OAXB-"Area of" ΔAOB`...........(1)

We know that  area of the sector `OAXB=θ/360xxpi r^2`

We also know that  area of `ΔAOB=r^2 sin  θ/2 cos  θ/2`  

Substituting these values in equation (1) we get,

`"Area of the segment AXB"=θ/360 xxpi r^2-r^2 sin  θ/2 cos  θ/2` 

`"Area of the segment AXB"=(θ/360 xxpi-sin  θ/2 cos  θ/2 )r^2`

`"So Area of the segment AXB"=((piθ)/360-sin  θ/2 cos  θ/2 )r^2` 

Therefore, area of the segment is` ((piθ)/360-sin  θ/2 cos  θ/2) r^2`