2009 AMC 10A Problems/Problem 6
A circle of radius is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
Area of the circle inscribed inside the semicircle Area of the larger circle (semicircle's area x 2) (4, or the diameter of the inscribed circle is the same thing as the radius of the semicircle). Thus, the area of the semicircle is Part of the semicircle that is unshaded is Therefore, the shaded part is
Thus the answer is
The solution above is the standard method of solving it, but we can also realize something about their radiuses. If we see, we know that the smaller circle has a radius of while the larger semicircle has a radius of . We know for a fact that if we have normal full circles and one has a radius that is double the other, the smaller circle will always be times smaller than the larger one (if you didn't know, now you know). Now, we have a semi-circle, the same thing applies but the larger circle is in half, so, therefore, the semi-circle (which is bigger than the smaller full circle), will be double the size of the small circle that is inscribed in it. Double means that of the small full circles will be able to fit the larger semi-circle. So, therefore, the shaded parts and the small full circle are the same, making parts, so the shaded parts are of the whole semi-circle
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