Formula for the Volume of a Octagon

In geometry, an octagon is a polygon with eight sides. A regular octagon has eight equal sides and equal angles. The regular octagon is commonly recognized from stop signs. An octahedron is an eight-sided polyhedron. A regular octahedron has eight triangles with edges of equal length. It is effectively two square pyramids meeting at their bases.

Octagon Area Formula

The formula for the area of a regular octagon with sides of length "a" is 2(1+sqrt(2))a^2, where "sqrt" indicates the square root.

Derivation

An octagon can be viewed as 4 rectangles, one square in the center and four isosceles triangles in the corners.

The square is of area a^2.

The triangles have sides a, a/sqrt(2) and a/sqrt(2), by the Pythagorean theorem. Therefore, each has an area of a^2/4.

The rectangles are of area a * a/sqrt(2).

The sum of these 9 areas is 2a^2 (1 + sqrt(2)).

Octahedron Volume Formula

The formula for the volume of a regular octahedron of sides "a" is a^3 * sqrt(2)/3.

Derivation

Regular Octahedron

The area of a four-sided pyramid is area of base * height / 3. The area of a regular octagon is therefore 2 * base * height / 3.

Base = a^2 trivially.

Pick two adjacent vertices, say "F" and "C." "O" is at the center. FOC is an isosceles right triangle with base "a," so OC and OF have length a/sqrt(2) by the Pythagorean theorem. So height = a/sqrt(2).

So the volume of a regular octahedron is 2 * (a^2) * a/sqrt(2) / 3 = a^3 * sqrt(2) / 3.

Surface Area

The regular octahedron's surface is the area of an equilateral triangle of side "a" times 8 faces.

To use the Pythagorean theorem, drop a line from the apex to the base. This creates two right triangles, with the hypotenuse of length "a" and one side length "a/2." Therefore, the third side must be sqrt[a^2 - a^2/4] = sqrt(3)a/2. So the area of an equilateral triangle is height * base/2 = sqrt(3)a/2 * a/2 = sqrt(3)a^2/4.

With 8 sides, the surface area of a regular octahedron is 2 * sqrt(3) * a^2.

References

About the Author

Paul Dohrman's academic background is in physics and economics. He has professional experience as an educator, mortgage consultant, and casualty actuary. His interests include development economics, technology-based charities, and angel investing.