a sector has a perimeter of 12cm and area 8 cm squared ; What is its radius?

Question

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Answer 1

A sector is a wedge (pie slice). The perimeter consists of 2 radii, and an arc length. You know the arc length is given by the equation:
s = r·Θ
So the perimeter is:
P = 2r + rΘ = r·(2 + Θ)
Where Θ is the angle describing the sector (Θ is in radians).
You also know that the area of a sector is given by the equation:
A = (Θ / 2)·r²

Now set up the system of equations:
1) r·(2 + Θ) = 12
2) (Θ / 2)·r² = 8

Solve (1) for Θ:
2 + Θ = 12/r
Θ = 12/r - 2

Now put this into (2):
[(12/r - 2) / 2]·r² = [12 / (2r) - 1]·r² = 6·r - r² = 8
──► r² - 6r + 8 = 0

Now use the quadratic formula to find r:
r = [6 ± √(36 - 32)] / 2 = (6 ± 2) / 2 = 3 ± 1 = {2, 4}

Evidently, r could be either 2 or 4.

Answer 2

P = r(2 + θ)
A = r^2θ/2
θ = 2A/r^2
P = 2r + 2A/r
2r^2 - Pr + 2A = 0
r = (P ± √(P^2 -16A))/4
r = 3 ± √(3^2 - 8)
r = 2, 4
θ = 4, 1

Answer 3

let x = % of total circle

8 = pi*r^2*x
x = 8/pi*r^2

12 = 2r + pi*2r*x
12 = 2r + pi*2r*(8/pi*r^2)
12 = 2r + 16/r
12r = 2r^2 + 16
2r^2-12r +16 = 0

Answer 4

C = phi * r
S = phi/2 * r^2

S / C = r/2 => r = 2S / C = 4/3