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# Scaling Circles

The area of circle $$A$$ is $$25\pi$$, and the area of circle $$B$$ is $$4\pi$$. What is ratio of the circumference of circle $$A$$ to the circumference of circle $$B$$.

(A) 2.5 : 1      (B) 5 : 2      (C) 5 : 4    (D) 25 : 16      (E) 125 : 1

Got it?

This problem is relatively straight forward, but there is a subtle concept lurking beneath the surface.

GOOD ENOUGH: The area of a circle is $$\pi(r^2)$$, so the radius of circle $$A$$must be 5 and the radius of circle $$B$$ must be 2. Just to be explicit:

$$\pi(r^2) = 25\pi$$

$$r^2 = 25$$

$$r = \sqrt25 = 5$$

The same algebra works for circle $$B$$.

Since the radius of circle $$A$$ is 5, and the radius  circle $$B$$ is 2, we kind find their circumferences using the formula $$C = 2\pi(r)$$. The circumference of circle $$A$$ is $$2\pi(5) = 10\pi$$. The circle of circumference of circle $$B$$ is $$2\pi(2) = 4\pi$$. The ratio of $$10\pi$$ to $$4\pi$$ simplifies:

$$10\pi : 4\pi = 10 : 4 = 5 : 2$$

In the problem we’re moving from area to circumference – from a property that is proportional to $r^2$ to a property that is proportional to $r$. In effect we are taking the square root of the original proportion.
$$\sqrt{25\pi} : \sqrt{4\pi} = 5\pi : 2\pi = 5 : 2$$