Problem 83

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Problem 83: (2021 Fall AMC 10b, 18; 12b 15)

Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$?

$(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147$

Solution

Exercise 83: Solution

Four identical equilateral-triangle sheets of paper, each with side length $6$, are stacked on top of each other. One sheet is rotated clockwise $30^\circ$ about its center, another by $60^\circ$, and a third by $90^\circ$. The resulting $24$-sided polygon is shown below. Its area can be expressed as $a\sqrt{b}-c$, where $a,b,c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c$?

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