Semicircle

Posts : 85 Join date : 2009-09-11

What is the area of the largest semicircle that can be inscribed in a unit square?

Posts : 28 Join date : 2009-10-20

Well, the largest semicircle should be half the radius of the largest circle and the radius of the largest circle is
half the diameter which is 1 so the radius of the largest circle is 0.5 and 0.5² is 0.25 so the area of
the largest circle is 0.25pi so the area of the largest semicircle is 0.125pi or pi/8

Posts : 85 Join date : 2009-09-11

Sadly, you've fallen into the trap! A larger semicircle would be obtained by this inscribed semicircle that is tangent to two sides.

Find the area of this one and compare to your answer.

Posts : 28 Join date : 2009-10-20

Spoiler:

Posts : 85 Join date : 2009-09-11

Congratulations, your final answer is correct! Smile

If your curious there is however one very small mistake in the presented form of your answer. The form of your final answer pi/(3+2sqrt2) would be considered incorrect to MATHCOUNTS graders. To simplify such fractions you can rationalize the denominator, since radicals are not allowed in the simplest form of a fraction.

AK wrote:: sqrt2/{sqrt2+1}. now it can't be simplified any longer

So in this case it can indeed be simplified by rationalizing the denominator. Multiply by (sqrt2-1) and that will eliminate the radicals in the denominator.