2022 AMC 12B Problems/Problem 25
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[hide]Problem
Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as , where , , and are integers and is not divisible by the square of any prime. What is ?
Solution
We calculate the area as the area of the red octagon minus the four purple congruent triangles: We first find the important angles in the figure. We note that 2 adjacent hexagons are rotated with respect to the other, so the angles between any sides is . In particular, as the purple triangles are isosceles, they have angles , and , and the octagon is equiangular (all its angles are ). Thus, we can draw a square around the octagon, and we note that the ``cut out" triangles are all isosceles right triangles.
Now, we calculate the side length of the square. Note that the hexagon has a height of , so the length of a side of the square is . In particular, the horizontal/vertical sides of the octagon have length , so the legs of the isosceles triangles are Thus, the area of the octagon is Now, we calculate the area of one of the four isosceles triangles. The base of the triangle is , so the area is Thus, the area of the dodecagon is Thus the answer is , or .
~cr. naman12
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last problem |
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All AMC 12 Problems and Solutions |
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