Question

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 $m + n + p?$

$-12$

$-4$

$4$

$24$

$32$

Solution:

Center the square at the origin with vertices $\left(\pm\tfrac12, \pm\tfrac12\right).$ 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 $\left(-\tfrac12, \sqrt3 - \tfrac12\right)$ to $\left(\tfrac12, \sqrt3 - \tfrac12\right).$

The outer boundary is a $12$ -gon with flat edges at distance $\sqrt3 - \tfrac12$ from the center, convex vertices such as $\left(\sqrt3 - \tfrac12, \tfrac12\right),$ and four reflex notches where adjacent hexagons' slanted edges meet, at $\left(\tfrac52 - \sqrt3, \tfrac52 - \sqrt3\right)$ and its symmetric images.

Applying the shoelace formula to these $12$ vertices gives area $16\sqrt3 - 23,$ so $m = 16,$ $n = 3,$ $p = -23,$ and $m + n + p = -4.$

Thus, the correct answer is B.

2022 AMC 12B Problem 25

Solution:

Problem 25 in Other Years