Kaustubh

Member • Dec 15, 2008

## What is the Size of the Largest Hexagon Inside Square?

Here's an easier one for you -

What is the size of the largest regular hexagon that can be constructed inside a square with side-length 's'?

Crack it fast, please.

**Solution:**

Problem Analysis: To determine the size of the largest regular hexagon that can be constructed inside a square with side-length 's', we need to find the maximum size of the regular hexagon while keeping it fully contained within the square.

Solution: A regular hexagon is a polygon with six equal sides and six equal angles. Let's analyze the problem step by step:

Determine the side length of the regular hexagon: Since the hexagon is regular, all six sides are equal in length. Let's denote the side length of the hexagon as 'h'.

Find the distance from the center of the hexagon to one of its vertices: The distance from the center of the hexagon to any of its vertices is equal to the side length of the hexagon (h).

Determine the distance from the center of the hexagon to one of its sides: The distance from the center of the hexagon to one of its sides is equal to the apothem (a). The apothem is the perpendicular distance from the center of the hexagon to one of its sides.

Calculate the length of the apothem: To find the apothem (a), we can divide the hexagon into two congruent triangles. Each of these triangles has a base equal to one side of the hexagon (h) and a height equal to the apothem (a). The angle between the base and the height in each triangle is 30 degrees, as the hexagon has interior angles of 120 degrees. Using trigonometry, we can calculate the apothem (a) as:

a = h * sin(30 degrees) = h * (1/2) = h / 2

Determine the side length of the square: Since the hexagon must be fully contained within the square, the distance from the center of the hexagon to one of its vertices (h) should be less than or equal to half the side length of the square (s/2).

Solve for the side length of the hexagon in terms of the square's side length: Using the previous step, we can set up the inequality:

h ≤ s/2

Calculate the maximum side length of the hexagon: By substituting a = h / 2 from step 4, we have:

h ≤ s/2 h/2 ≤ s/4 a ≤ s/4

So, the maximum length of the apothem (a) is equal to s/4. Since the distance from the center to the vertex is twice the apothem, the maximum side length of the hexagon is 2 * (s/4), which simplifies to s/2.

Example: Let's take a square with a side length of 10 units (s = 10). To find the size of the largest regular hexagon that can be constructed inside this square, we can use the formula derived above:

Maximum side length of hexagon = s/2 = 10/2 = 5 units

Therefore, the largest regular hexagon that can be constructed inside a square with a side length of 10 units has a side length of 5 units.

Note: The same approach can be applied to squares of different sizes by substituting the corresponding value of 's' in the formula.