Problem 126

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Problem 126: (2022 AMC 12b, 25)

Four regular hexagons surround a square with a 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 $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?

$\textbf{(A) } -12 \qquad$ $ \textbf{(B) }-4 \qquad$ $ \textbf{(C) } 4 \qquad$ $ \textbf{(D) }24 \qquad$ $ \textbf{(E) }32$

Solution

Exercise 126: Solution

Three squares surround an equilateral triangle with a side length $2$, each one sharing an edge with the triangle, as shown in the figure below. The area of the resulting 9-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?

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