how do you find the arc length when you are given the radius and area in terms of pi

Question

the radius is 5cm 

the area of the sector is 5(pi)cm^2

the centre angle is not given and the questions asks for the answer to be left in terms of pi

 

Answer 1

The arc length should be in the same proportion to the circumference of the circle as the area subtended by the arc is to the area of the complete circle.

 

Area subtended by arc = 5 pi

 

Area of circle =pi * r^2 = 25 pi because r = 5 cm

 

Ratio of areas is therefore 5 pi/25 pi = 1/5

 

The circumference of the circle with radius of 5 is 2*pi*r = 10 pi

 

The length of the arc is therefore 1/5 (10 pi) = 2 pi

 

 

Answer 2

Area of sector:

 

A = (0.5)(r²)θ        substitute values

5π = 0.5(5²)θ         multiply both sides by 2

10π = 25θ             divide both sides by 25

10π/25 = θ            reduce

2π/5 = θ