Question

If a central angle of 90 degrees defines an arc on circle R that has the same length as the arc on circle W defined by a 60-degree central angle, what is the ratio of the area of circle R to the area of circle W? Express your answer as a common fraction.

Answer 1

let the radius of circle R be r
let the radius of circle W be w

recall that arclength = rØ, where Ø is the angle in radians
arc length on first circle = (π/2)r
arc length of 2nd circle = (π/3)w
but they are equal
πr/2 = πw/3
r = 3w/2

area of first sector = (1/4)πr^2
= (1/4)π(9w^2)/4
= (9/16)πw^2

area of 2nd sector = (1/6)π w^2

ratio of first area : second area
= (9/16)π w^2 : (1/6)π w^2
= 9/16 : 1/6
= 54 : 16
= 27 : 8

or 27/8 as a fraction

check my arithmetic

Answer 2

since the arc lengths are the same,

90r = 60w
So, r/w = 2/3

Since the areas vary as the square of the radii,

R/W = = (r/w)^2 = 4/9

Answer 3

Steve is right,
I can't be bothered to find my error in all my mess.

Answer 4

360 / 60 = 6

Answer 5

Yes