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.
TL;DR (Too Long; Didn't Read)
The simplest, and by far most common, way of finding the length of a regular hexagon's sides is using the following formula:
s = P ÷ 6, where P is the perimeter of the hexagon, and s is the length of any one of its sides.
Calculating Hexagon Sides From the Perimeter
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:
48 inches ÷ 6 = 8 inches.
Each side of your hexagon measures 8 inches in length.
Calculating Hexagon Sides From the Area
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 × √3) × s2, 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:
Substitute Area Into the Equation
Start by substituting the area of the hexagon into the equation:
128 = (1.5 × √3) × s2
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:
128 ÷ (1.5 × √3) = s2
Conventionally the variable goes on the left side of the equation, so you can also write this as:
s2 = 128 ÷ (1.5 × √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:
s2 = 128 ÷ (1.5 × 1.732)
Which simplifies to:
s2 = 128 ÷ 2.598
Which, in turn, simplies to:
s2 = 49.269
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:
√s2 = √49.269 then becomes:
s = 7.019 inches