Chapter 1: Introduction to the Problem

Can you determine the area of a square that fits perfectly inside a hexagon?

This geometry challenge, derived from SMC 2023 Q24, invites you to solve for the area of a square nestled within a regular hexagon with a side length of 1. This question was posed to students aged 14 to 18 aiming to qualify for the BMO1 in the UK. Do you possess the skills needed to tackle this challenge? Let’s find out!

To start, we’ll focus on one corner of the hexagon, effectively “zooming in” on this particular area:

If we can ascertain the side length of the square, we will have effectively solved the problem. Let’s redefine our labels: assume the side length of the square is represented as 2x. Consequently, we can express our variables since we are effectively halving the square. We have the equation a + b = 1, as the hexagon’s side length is 1.

We recognize that the sum of the interior angles of a hexagon is 720 degrees, meaning each angle measures 120 degrees. Since we are bisecting one of these angles, we can label the following:

Using trigonometric principles, we determine that x = a sin(60) = a sqrt(3)/2. Additionally, since the upper portion must equal 1/2, we have x = b/2 + 1/2.

At this point, we recall that a + b = 1. What if we could express x in terms of a + b? Rearranging leads us to 2x - 1 = b, and also 2x/sqrt(3) = a. Thus, we find that a + b = (x - 1) + 2x/sqrt(3). Since a + b is equal to 1, we can set up a linear equation for x.

From here, solving for x yields x = (3 - sqrt(3))/2, leading us to conclude that the side length of the square, 2x, is equal to 3 - sqrt(3). Squaring this value provides us with the area, resulting in 12 - 6sqrt(3).

Chapter 2: Visual Aids and Further Exploration

To enhance your understanding of this geometrical concept, take a look at the following videos:

The first video explains how to find the area of a circumscribed hexagon given the area of an inscribed hexagon.

The second video tackles the area of a hexagon inscribed within a circle, providing additional insights into this geometry problem.

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