Arc Length and Sector of Area│Please Help!

Question

Fill in the table:

Radius = ?

Central Angle = ?

Arc Length = pi / 2

Sector Area = pi / 8

Answer 1

Arc Length (S) equals the radius (r) multiplied by the angle (θ) where θ is in radian units.

S = rθ

π/2 = rθ

or (solving for "θ")

θ = π/(2r)

Sector Area (A_S) equals π multiplied by the radius squared multiplied by the ratio of the angle (θ), in radians) divided by 2π

A_S = πr²(θ/(2π))

π/8 = πr²(θ/(2π))

1/8 = r²(θ/(2π))

2π/8 = r²θ

π/4 = r²θ

plugging in (from the first equation) "π/(2r)" in place of "θ"

π/4 = r²(π/(2r))

1/4 = r/2

r = 2/4 = 1/2

Plugging in r = 1/2 into θ = π/(2r) we get:

θ = π/(2(1/2))

θ = π