Formula for the Volume of a Hexagon

Hexagon and parallelogram.

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.

Derivation

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.

Alternate Derivation

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.

Related Articles

Formula for the Volume of a Octagon
How to Calculate Height From Volume
How to Find the Volume of a Triangular Pyramid
How to Convert Square Dimensions to Round
How to Find the Area of Triangles & Trapezoids
How to Find the Volume of a Semicircle
How to Calculate Volume of a Circular Cylinder
How to Calculate Area of an Obtuse Triangle
How to Calculate the Area of a Curved Surface
How to Find the Height of a Rhombus
How to Calculate Perimeter and Area Ratio
How to Find the Radius of a Cone
How to Find Degrees in Polygons
How to Calculate Lateral Area
How to Solve a Hexagon
How to Calculate the Base of a Shape
How to Calculate the Perimeter of a Diamond
How to Calculate Surface Area From Volume
How to Calculate the Square Foot of an Irregular Polygon
How to Calculate Volume in Cubic Centimeters

Dont Go!

We Have More Great Sciencing Articles!