Problem 20 · 2013 AMC 8
Medium
Geometry & Measurement
inscribed-rectanglepythagorean
A 1 × 2 rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
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Answer: C — π.
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Hint 1 of 2
By symmetry, the center of the semicircle sits at the middle of the diameter — right below the rectangle's center. A radius drawn to an upper corner is the hypotenuse of a little right triangle. What are its two legs?
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Hint 2 of 2
Don't chase the radius directly — build a right triangle from the center to a point on the circle and use the Pythagorean theorem. Also note area only needs r2, so you never have to simplify √2.
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Approach: right triangle from the center to a top corner
- Put the center at the midpoint of the diameter. The rectangle (long side 2 on the diameter, height 1) reaches a top corner that is 1 across and 1 up from the center.
- That corner lies on the circle, so the radius is its distance: r2 = 12 + 12 = 2. (No need to take the square root.)
- Semicircle area = ½πr2 = ½π(2) = π.
- Worth keeping: area formulas use r2, so stop the moment you know r2 — squaring back a messy radius is wasted work.
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