2020 · Q86
What is the area of a segment of a circle of radius r subtending an angle θ at the centre?
Elementary MathematicsCircles
- A(1/2)r²θ
- B(1/2)r²(θ − 2sin(θ/2)cos(θ/2))
- C(1/2)r²(θ − sin(θ/2)cos(θ/2))
- D(1/2)r²sin(θ/2)cos(θ/2)
Explanation
Segment area = sector minus triangle = (1/2)r^2*theta - (1/2)r^2*sin(theta) = (1/2)r^2(theta - 2sin(theta/2)cos(theta/2)). Hence option (B). Since 2sin(theta/2)cos(theta/2) = sin(theta), this is the familiar (1/2)r^2(theta-sin(theta)).