Question

The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle of the sector in degrees, minutes and seconds

Answer 1

Let OAB be the sector of a circle of radius r.

The angle of the sector is θ.

Perimeter of the sector = OA + arc AB + OB arc AB = rθ

∴ Perimeter of the sector = r + r θ + r

= 2r + rθ

= r(2 + θ)   ......(1)

Length of the arc of the semi – circle of radius

l = nπ     ......(2)

Given that perimeter the circular sector = Length of the arc of the semi circle of radius r

From equations (1) and (2), we have

r(2 + θ) = πr

2 + θ = π

θ = π – 2

θ = `22/7 - 2 = (22 - 14)/7`

θ = `8/7` radians

θ = `(8/7 xx 180/pi)^circ`

= `8/7 xx 180/22 xx 7`

θ = `720/11`

= 65° 27' 16"