We can find the area of a regular hexagon by splitting it into 6 equilateral triangles
L is the side length. H is the height of each triangle.
Using Pythagoras' theorem for a right angle triangle:
Now, looking at one of the equilateral triangles:
In approximate numeric terms, the area of a regular hexagon is 2.598 times the square of its side length.
If you know the smallest width W of the hexagon
the formula is:
In approximate numeric terms, the area of a regular hexagon is 0.866 times the square of its smallest width.
Click here for the areas of some particular hexagons, and of hexagons which tile or tessellate
In general, if your hexagon is not regular (for instance if its sides are not the same length), and if it does not tessellate, it may be easiest to cut it into convenient triangles, squares or rectangles and calculate the area of each of them.
Maximum hexagon area
If you have a hexagon with known side lengths, it will have the maximum possible area if all its vertices lie on a circle. Such a hexagon is known as 'cyclic hexagon'.
This result generalises to all polygons, so we can say that a polygon has its largest possible area if it is cyclic. The result is rather elegantly proved here.
Area by computer program
If you are writing a computer program, try the comp.graphics.algorithms faq, available from http://www.faqs.org/ among other places. At the time of writing http://geometryalgorithms.com/Archive/algorithm_0101/ has a standard and simple algorithm for calculating polygon area. For convenience, the code is also in a file here.
page date: 25Oct05. I enjoy correspondence stimulated by this site. You can contact me here.