Area of circle given arc length?

Question

number 50?

Answer 1

Answer : Option C is correct.

It can be clearly shown that --

The two circles have the same radius (r) and PS = ST = PR = RT

Also M is midpoint of RS so that MS = MR = r/2

Hence cos θ = MS/PS = (r/2) / r  =  1/2

=> θ = 60° , therefore,  Angle PST =  120°

Consider circle with center S :

Arc PRT subtends 120° at the center. 

Hence its length PRT ( given = 2 π )  =  (120/360) * 2 π r

=> r/3  =  1 

=>  r  =  3 

Hence area of the circle =  π (3)² =  9 π  ...................... Answer

Answer 2

PRS and RST have sides that are all equal to the radius, so they are both equilateral triangles. 

The angles of an equilateral triangle are all 60° so angle PRT is 120° and the arc PRT is equal to 1/3 of a full circle.

The circumference of the full circle will be 6π which means the radius is 3.

C = 2πr = 6π --> r = 3

From that you can easily find the area:

A = πr²

A = 3² * π

A = 9π

Answer:

(C) 9π