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2018 AMC 12B Problem 11

Problem 11 of 25IntermediateGeometry

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point AA in the figure on the right. The box has base length ww and height h.h. What is the area of the sheet of wrapping paper?

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Solution

Following a fold from a corner of the paper to the center of the box top, the distance from a corner of the sheet to its center is w2+h+w2=w+h. \dfrac{w}{2}+h+\dfrac{w}{2}=w+h. That segment is a leg of a 4545-4545-9090 triangle whose hypotenuse is a full side of the square sheet, so the side length is 2 (w+h).\sqrt2\,(w+h). The area of the sheet is (2 (w+h))2=2(w+h)2.\left(\sqrt2\,(w+h)\right)^2=2(w+h)^2. Thus, the correct answer is A.

More practice

Concepts: special right triangle · area

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