Problem 16 · 2010 AMC 8
Easy
Geometry & Measurement
area-equation
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
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Answer: B — √π.
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Hint 1 of 2
You want the ratio s/r, and the only fact is ‘same area.’ Write both areas, set them equal, and notice s/r appears once you divide — squared.
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Hint 2 of 2
To get a length ratio from an area condition, expect a square root: areas compare like (length ratio)2, so undo it by taking √.
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Approach: equate areas, then take a root
- Equal areas: s2 = πr2. Divide both sides by r2 to isolate the ratio: (s/r)2 = π.
- Take the square root: s/r = √π.
- Sanity check: π > 1, so √π > 1 — the square's side should beat the circle's radius, which feels right since a circle of radius r is wider than r across. Answers like π or π2 skip the square root that the area-to-length step demands.
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