Question

Suppose that circles R and S have a central angle measuring 80°. Additionally, the measure of the sector for circle R is 32 9 π m2 and for circle S is 18π m2. If the radius of circle R is 4 m, what is the radius of circle S?

Answer 1

We know that

The measure of the sector of circle R is 32π/9 m².
The measure of the central angle is 80°. 
This means that the sector is 80/360 = 2/9 of the circle. 
The area of a circle is given by A=πr²,
so
 the area of the sector is A=2/9πr².
  To verify this, 2/9π(4²) = 2/9π(16) = 32π/9.

Using this same formula for circle S, we will work backward to find the radius:
18π = 2/9πr²
Multiply both sides by 9:18*9π = 2πr²162π = 2πr²
Divide both sides by 2π:162π/2π = 2πr²/2π81 = r²
Take the square root of both sides:√81 = √r²  r=9 m

the answer is 
the radius of circle S is r=9 m

alternative method

Let 

rA--------> radius of the circle R

rB-------> radius of the circle S

SA------> the area of the sector for circle R

SB------> the area of the sector for circle S

 

we have that

rA=4 m

rB=?

SA=32π/9 m²

SB=18π m²

 

we know that

 

if Both circle A and circle B have a central angle , the square of the ratio of the radius of circle A to the radius of circle B is equals to the ratio of the area of the sector for circle A to the area of the sector for circle B

 (rA/rB) ^2=SA/SB-----> rB²=(SB/SA)*rA²-----> rB²=(18π/32π/9)*4²-----> rB²=162/2

rB=9 m