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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:

\(10\pi : 4\pi = 10 : 4 = 5 : 2\)

The answer is (B)

800: This is an example of scaling. We can scale a geometric figure by multiply any dimension by a constant. The constant (aka “scaling factor”), will not affect all the properties in the figure in the same way.That is, if we multiply the radius of a circle by 3, then the radius is 3 times as long as it was before, but the area of the circle is 9 times as large as it was before. This is because the area of a circle is directly proportional to the square of it’s radius.

In the problem we’re moving from area to circumference – from a property that is proportional to [math]r^2[/math] to a property that is proportional to [math]r[/math]. In effect we are taking the square root of the original proportion.

\(\sqrt{25\pi} : \sqrt{4\pi} = 5\pi : 2\pi = 5 : 2\)