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2006 AMC 10B Problem 6

Problem 6 of 25EasierGeometry

A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure 2π,\tfrac{2}{\pi}, as shown. What is the perimeter of this region?

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Solution

Each side has length 2π,\tfrac{2}{\pi}, the diameter of a semicircular arc, so each arc has length 12π⋅2π=1.\tfrac12\pi\cdot\tfrac{2}{\pi}=1. The boundary consists of four such arcs, so the perimeter is 4⋅1=4.4\cdot1=4. Thus, the correct answer is D.

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Concepts: arc · perimeter · circumference

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.