In geometry, a hexagon is a polygon with six sides. A regular hexagon has six equal sides and equal angles. The regular hexagon is commonly recognized from honeycomb and the interior of the Star of David. An hexahedron is a six-sided polyhedron. A regular hexahedron has six triangles with edges of equal length. In other words, it is a cube.
Hexagon Area Formula
The formula for the area of a regular hexagon with sides of length "a" is 3 --- sqrt(3) --- a^2 / 2, where "sqrt" indicates the square root.
A regular hexagon can be viewed as six equilateral triangles of sides a. Their angles are 60 degrees, so the angles in the hexagon are 120 degrees. The triangles can be extended below the hexagon to form a parallelogram of sides 2a. A larger triangle can be created to determine the height of this parallelogram, which is 2a --- cos 30° = a --- sqrt(3).
The parallelogram in the figure is therefore of area height --- base = (a --- sqrt(3)) --- 2a = 2 --- sqrt(3) --- a^2.
But this is for a parallelogram made up of 8 equilateral triangles. The hexagon was only composed of 6. So the hexagon's area is 0.75 of this, or 3 --- sqrt(3) --- a^2 / 2.
The six equilateral triangles in a hexagon have sides "a." Their heights, h, are, by the Pythagorean theorem, sqrt[a^2 - (a/2)^2] = a --- sqrt(3) / 2.
The area of a triangle is therefore (½) --- base --- height = (a) --- [a --- sqrt(3) / 4]. Six triangles in the hexagon give an area of 3 --- sqrt(3) --- a^2 / 2.
Hexahedron Volume Formula
The formula for the volume of a regular hexahedron of sides "a" is a^3, since a regular hexahedron is a cube.
The surface area is, of course, a^2 --- 6 sides = 6a^2.