2022 AMC 12B Problem 25

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

Answer: B

Concepts:regular polygon coordinate geometry shoelace formula

Difficulty rating: 2520

Small hint:

Each hexagon shares an edge with the square and reaches across to the far side; set up coordinates with the square centered at the origin

Big hint:

The outer boundary has flat edges at distance

\sqrt3 - \tfrac12

and reflex notches where adjacent hexagons’ edges cross, at

\left(\tfrac52 - \sqrt3,\ \tfrac52 - \sqrt3\right)

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.