SOLUTION: The diameter of the large circle is twice the diameter of the small circle. What is the ratio of their areas? A) 1:2 B) 1:3 C) 1:4 D) 1:5 E) 1:6

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Question 252787: The diameter of the large circle is twice the diameter of the small circle. What is the ratio of their areas?
A) 1:2 B) 1:3 C) 1:4 D) 1:5 E) 1:6
Answer by Theo(13342) About Me  (Show Source):
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the radius of a circle is equal to (1/2) * the diameter of the circle.

let the diameter of the smaller circle = 2 * r
this makes the radius of the smaller circle = r

since the diameter of the larger circle is 2 times the diameter of the smallet circle:

this makes the diameter of the larger circle = 4 * r
this makes the radius of the larger circle = 2 * r

area of the smaller circle = 2 * pi * r^2
area of the larger circle = 2 * pi * (2 * r)^2 = 2 * pi * 4 * r^2 = 8 * pi * r^2

area of smaller circle / area of larger circle becomes:

2 * pi * r^2 / 8 * pi * r^2

pi * r^2 cancel out so you get 2 / 8 which simplified to 1 / 4.

ratio of area of smaller circle to larger circle is 1 / 4 = 1:4.

answer is selection C (1:4).

example:

let diameter of smaller circle = 6.
radius of smaller circle is 3 and area of smaller circle is 2 * pi * 3^2 = 2 * pi * 9 = 56.54866776

diameter of larger circle = 2 * 6 = 12
radius of larger circle is 6 and area of larger circle is 2 * pi * 6^2 = 2 * pi * 36 = 226.1946711

ratio of area of smaller circle to area of larger circle equals:

56.54866775/226.1946711 = .25 = 1/4.