How to find the area of a regular hexagon from its side length

Key concept: Regular hexagons have a nice geometry because they can be split into 6 equilateral triangles. That makes exact area expressions much easier than they first look.

Step by step

A regular hexagon can be divided into 6 congruent equilateral triangles.

Step 1: Use the area formula for a regular hexagon

The area of a regular hexagon with side length $s$ is

$A=\frac{3\sqrt{3}}{2}s^2$

Step 2: Substitute $s=48$ mm

$A=\frac{3\sqrt{3}}{2}(48)^2$

Step 3: Simplify

$48^2=2304$

so

$A=\frac{3\sqrt{3}}{2}\cdot 2304$

$A=3456\sqrt{3}$

Final answer

$\boxed{3456\sqrt{3}\text{ mm}^2}$

That is the exact area of the regular hexagon.

Pitfall alert

Do not approximate $\sqrt{3}$ if the question asks for exact form. Another common mistake is forgetting to square the side length before multiplying by the constant factor. Also, be careful with units: the side length is in millimeters, so the area must be in square millimeters.

Try different conditions

If the side length were different, the same formula still works: just replace $48$ with the new value. If you were asked for a decimal approximation instead of exact form, you could evaluate $3456\sqrt{3}$ with a calculator and round at the end.

Further reading

regular hexagon, exact area, equilateral triangle