Flower Petals (Fall 2021 AMC 12B #15)

Fall 2021 AMC 12B #15

Problem: 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 $a+b+c$?

Solution: We can see that the resulting shape is a "regular" 24-sided polygon with equal side lengths, and equal interior angles. So we can take a "petal" of the polygon that consists of 3 adjacent vertices, find the area, and multiply by 12. The key here is to find the area of the petal by subtracting areas from a corner of the original square paper.

From this diagram of a petal inside a $3 \times 3$ corner of a square, we can see that each petal has an area of $3 \cdot 3 - 2(\frac{1}{2}(3)(\sqrt3)) = 9 - 3\sqrt3$. So, we get that the area of the polygon is $12 \cdot (9-3\sqrt3) = 108 - 36\sqrt3$, which gives us an answer of $108+36+3 = 147$.

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