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2022 AMC 12B Problem 25

Problem 25 of 25HarderGeometry

Four regular hexagons surround a square with a side length 1,1, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 1212-sided outer nonconvex polygon can be written as mn+p,m\sqrt n + p, where m,m, n,n, and pp are integers and nn is not divisible by the square of any prime. What is m+n+p?m + n + p?

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Solution

Center the square at the origin with vertices (±12,±12).\left(\pm\tfrac12, \pm\tfrac12\right). Each hexagon shares one edge with the square and extends across to the opposite side; the hexagon on the bottom edge, for instance, has its far (top) edge from (−12,3−12)\left(-\tfrac12, \sqrt3 - \tfrac12\right) to (12,3−12).\left(\tfrac12, \sqrt3 - \tfrac12\right). The outer boundary is a 1212-gon with flat edges at distance 3−12\sqrt3 - \tfrac12 from the center, convex vertices such as (3−12,12),\left(\sqrt3 - \tfrac12, \tfrac12\right), and four reflex notches where adjacent hexagons’ slanted edges meet, at (52−3,52−3)\left(\tfrac52 - \sqrt3, \tfrac52 - \sqrt3\right) and its symmetric images. Applying the shoelace formula to these 1212 vertices gives area 163−23,16\sqrt3 - 23, so m=16,m = 16, n=3,n = 3, p=−23,p = -23, and m+n+p=−4.m + n + p = -4. Thus, the correct answer is B.
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Tagged: regular polygon · coordinate geometry · shoelace formula

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