how to find the radius and angle of the sector?

Question

if given the perimeter and area of the sector are 19 cm and 22.5 cm repectively.

Answer 1

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.

Answer 2

Subtract twice the radius from the perimeter and you have the fraction of the circumference that is a part of the sector.

Divide the area by pi * r^2 and you have the fraction of the circle's area taken up by the sector.

Combine the two with a little algebra and you have your answer.

Answer 3

2pi r=19
pi r^2=22.5
r^2=19/3.14=6.05
r=sqrt(6.05)
angle of the sector=(angle/360 degrees) xpi r^2