Circle I is circumscribed about a given square and circle II is inscribed in the given square. If r is the ratio of the area of circle I to that of circle II, then r equals:

1966 AMC 12 Problem 4

Attempt Problem 4 of the 1966 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1966 AMC 12 solutions, or check the answer key.

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Circle

\mathrm{I}

is circumscribed about a given square and circle

\mathrm{II}

is inscribed in the given square. If

r

is the ratio of the area of circle

\mathrm{I}

to that of circle

\mathrm{II},

then

r

equals:

\sqrt2

2

\sqrt3

2\sqrt2

2\sqrt3

For square side

s,

compare radii

\frac{s}{\sqrt2}

and

\frac{s}{2}

Circle areas are proportional to the squares of their radii

For square side

s,

the outer circle has radius

\frac{s}{\sqrt2},

while the inner circle has radius

\frac{s}{2}.

Hence

r=\frac{\pi(\frac{s}{\sqrt2})^2}{\pi(\frac{s}{2})^2}=2.

Therefore, the correct answer is B.