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I'm working on a question from an old Mu Alpha Theta exam. I'm stuck and would appreciate any hints. (Please no answers! I'm sure I'm just missing something stupid. Also, this is for fun. Not HW.)

A square of area 8 is inscribed in a semi-circle with radius r. What is the area of a square inscribed in a circle with radius r?

My diagram has both squares in the same circle. I have the diagonal of the larger square as the diameter of the semi-circle containing the smaller square. Also, I am aware of the fact that if two figures are similar by the ratio k, that the ratio k^2 is the ratio of their areas.

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2 Answers

up vote 1 down vote accepted

Draw a radius of the circle from one of the vertices of the smaller square to the center of the circle. This will create a right triangle that will let you determine the circle's radius.

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Thank you. Today was not my best day... I knew I was missing something dumb. –  Josh Infiesto Feb 4 '13 at 7:04
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Consider this image, depicting the smaller square subdivided into quadrants:

enter image description here

$r$ is the circumradius of that smaller square; $R$, the radius of the semicircle, becomes the circumradius of the larger square. Thus,

$$\frac{\text{area of larger square}}{\text{area of smaller square}} = \frac{R^2}{r^2}$$

The ratio on the right-hand side is easy to determine.

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