Best Answer:
P(sector) = 2r + (a/2 pi) C = 2r + (a/2 pi)(2 pi r) = r(a + 2)

where r is radius, a is angle (radians), C is circle circumference

A(sector) = (a/2 pi) A(circle) = (a/2 pi)(pi r^2) = (a/2) r^2

Using the above equations, solve for r and a:

a + 2 = P/r ==> a = P/r - 2

a = 2A/r^2

P/r - 2 = 2A/r^2

Pr - 2r^2 = 2A

2r^2 - Pr + 2A = 0

r^2 - Pr/2 = -A

r^2 - Pr/2 + (P/4)^2 = (P/4)^2 - A (completing the square)

(r - P/4)^2 = (P^2 - 16A)/16

r = P/4 +/- sqrt(P^2 - 16A)/4

r = [P +/- sqrt(P^2 - 16A)] / 4

This gives the radius. Substitute the radius value into either of the above angle formulas to get the other piece of your answer.

Now let's plug in numbers to see what happens. For P = 19 and A = 22.5:

r = [P +/- sqrt(P^2 - 16A)] / 4

r = [19 +/- sqrt(361 - 360)] / 4 = (19 +/- 1)/4

Surprise! There are two roots of the quadratic: r = 5 and r = 4.5. Now let's get the angle, using a = 2A/r^2:

For r = 5, a = 45/25 = 1.8 radians

For r = 4.5, a = 45/(4.5^2) = 10/4.5 = 100/45 = 2 2/9 radians = 2.222... radians

So you have your choice of two equally valid solutions. The radius is 5 cm and the angle is 1.8 radians, or else the radius is 4.5 cm and the radius is 2 1/9 radians.

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