How To Calculate Length Of Sides In Regular Hexagons

The six-sided hexagon shape pops up in some unlikely places: the cells of honeycombs, the shapes soap bubbles make when they're smashed together, the outer edge of bolts, and even the hexagon-shaped basalt columns of the Giant's Causeway, a natural rock formation on the north coast of Ireland. Assuming you're dealing with a regular hexagon, which means all its sides are of the same length, you can use the hexagon's perimeter or its area to find the length of its sides.

Because a regular hexagon has six sides of the same length, finding the length of any one side is as simple as dividing the hexagon's perimeter by 6. So if your hexagon has a perimeter of 48 inches, you have:

\(\frac{48 \text{ inches}}{6} = 8 \text{ inches}\)

Each side of your hexagon measures 8 inches in length.

Just like squares, triangles, circles and other geometric shapes you may have dealt with, there is a standard formula for calculating the area of a regular hexagon. It is:

\(A = (1.5 × \sqrt{3}) × s^2\)

where ​A​ is the hexagon's area and ​s​ is the length of any one of its sides.

Obviously, you can use the length of the hexagon's sides to calculate the area. But if you know the hexagon's area, you can use the same formula to find the length of its sides instead. Consider a hexagon that has an area of 128 in²:

1. Substitute Area Into the Equation

Start by substituting the area of the hexagon into the equation:

\(128 = (1.5 × \sqrt{3}) × s^2\)

2. Isolate the Variable

The first step in solving for ​s​ is to isolate it on one side of the equation. In this case, dividing both sides of the equation by (1.5 × √3) gives you:

\(\frac{128}{1.5 × \sqrt{3}} = s^2\)

Conventionally the variable goes on the left side of the equation, so you can also write this as:

\(s^2=\frac{128}{1.5 × \sqrt{3}}\)

3. Simplify the Term on the Right

Simplify the term on the right. Your teacher might let you approximate √3 as 1.732, in which case you'd have:

\(s^2=\frac{128}{1.5 × 1.732}\)

Which simplifies to:

\(s^2=\frac{128}{2.598}\)

Which, in turn, simplies to:

\(s^2 = 49.269\)

4. Take the Square Root of Both Sides

You can probably tell, by examination, that ​s​ is going to be close to 7 (because 7² = 49, which is very close to the equation you're dealing with). But taking the square root of both sides with a calculator will give you a more exact answer. Don't forget to write in your units of measure, too:

\(\sqrt{s^2} = \sqrt{49.269}\)

then becomes:

\(s = 7.019 \text{ inches}\)