ShmoopTube

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Here’s your shmoop du jour, brought to you by radius envy.

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As observed by the renowned psychoanalyst Segment Freud.

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If the large circle has a center of H and the small circle has a center of I,

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and K is the midpoint of IH, what percent of the large circle (Circle H) is shaded?

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And here are the potential answers...

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We’ve got circles and triangles and intersections and midpoints up the wazoo.

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That shaded area is what we’re after…

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it’s both part of a triangle and part of both circles, so…where to start?

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Well, what are we given? We’re told that segment KI is half the length of HI.

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We aren’t given any numbers, so… heck, let’s just throw some in here.

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If we call KI “1,” then HI can be “2.”

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We’re also told that I is the midpoint of the smaller circle, so we know KI is the radius

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of that smaller circle. So is IL…which makes that segment also equal to “1.”

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Now…what about the radii of the larger circle?

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We’ve already determined that HI is 2… so HJ, another radius, would also be 2.

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And we know that angle IHJ is 90 degrees.

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Well, now we’ve got a right triangle with 2 sides of equal length…meaning we have an

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isosceles triangle on our hands…

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…which, by definition, has two angles of 45 degrees.

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All right, so angle KLI is 45 degrees.

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A full circle is 360 degrees… so the area of the shaded section is 45/360ths of the smaller circle.

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45 divided by 360 is 0.125…or 12.5% of the area of the small circle.

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12.5% is one of our answer choices… but don’t let ‘em rope you in here.

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We’re looking for the percentage of the larger circle.

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The area of the small circle…or, of any circle, for that matter, is πr squared.

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We’ve decided to call our radius “1,” so our area in this case is π(1)2, or just…pi.

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The shaded area is therefore 12.5% of pi, or 0.125 pi.

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So…what percentage of the larger circle is that?

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Well, we need to do the same thing we just did to find the area of the big guy.

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Here, our radius is 2, so the area is pi times 2 squared… or 4 pi.

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So finally, to find the percentage, we take our 0.125 pi divided by 4 pi…

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…to get 0.03125…

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…or 3.125%.

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Answer B.

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That’s a long way to go just to find a tiny shady area.

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