A circle has a sector with area 45/4pi and central angle of 9/10pi radians

Question

Answer in pi or decimal

Answer 1

Are you looking for the radius? The diameter? The arc length?

Sector Area = 0.5 r² θ

45/4 pi = 1/2 r² 9/10 pi

divide both sides by pi

45/4 = 1/2 r² 9/10

Multiply both sides by 20 to get rid of all denominators

20 ·45/4 = 9 r²

5 · 45 = 9 r²

Divide both sides by 9

5 ·5 = r²

So r = 5

Diameter = 10

Arc Length = r θ = 5 (9/10) pi = 4.5 pi

Answer 2

I think your question is, "What is the radius of the circle?" I am not sure what else it would be.

For a circle of radius r, the area of a sector of angle θ (in radians) is A = [θ/(2π)] πr². To interpret this, we know the area of the circle itself is πr² and the sector forms a fraction of θ/(2π) of that area since there are 2π radians in a circle. The π's in the area formula cancel out and the formula for the area of a sector reduces to

A = (1/2)r²θ. We are given the area A and the angle θ. You can solve the area formula for r.

r = [2A/θ]^1/2

where the 1/2 power indicates a square root of 2A/θ.

The way you write your area and angles are confusing. I think you mean to say that the given area is (45/4)π and that the angle is (9/10)π radians. If so, then

r ={(2)(45/4)π / [(9/10)π]}^1/2

We can cancel out the π's. We then have

r = [(45/2) / (9/10)]^1/2

r = [(45/2)(10/9)]^1/2

r = [(5)(5)]^1/2

r = 5