Surface Area and Volume of an Octahedron | Formula and Examples
A regular octahedron is one of the five Platonic solids. As a regular polyhedron, all of its faces are equal, and each vertex has the same degree. On a regular octahedron there are eight equilateral triangle faces, six vertices, and 12 edges. If you know the edge length of an octahedron, you can easily compute the volume and surface area with two simple geometric formulas. The surface area formula is simply the sum of the areas of the eight triangular faces. The volume formula can be derived in several different ways.
It is worth pointing out that "octahedron" technically means any polyhedron with eight faces. There are many solid shapes that fall into the category of octahedron; here we examine only the regular octahedron, the shape that most people think of when they hear "octahedron." Some specimens are shown below.
Related to the octahedron is the set of octahedral numbers. Octahedral numbers are those that can be represented as spheres packed into an octahedral shape. These are discussed after the examples.
Other Solids in the Family of Octahedra
Other 8-faced solid geometric figures include the triangular cupola (4 triangles, 3 rectangles, 1 hexagon), the hexagonal prism (6 rectangles, 2 hexagons), the truncated tetrahedron (4 triangles, 4 hexagons), the heptagonal pyramid (7 triangles, 1 heptagon), and the gyrobifastigium (4 triangles, 4 rectangles).
Surface Area of an Octahedron
If an octahedron has an edge length of L, then each of the equilateral triangular faces has an area of [sqrt(3)/4]*L^2. Since there are eight of them, we get the surface area formula
S = 8*[sqrt(3)/4]*L^2
S = 2*sqrt(3)*L^2
For example, if an octahedron has an edge length of 5 cm, its surface area is 50*sqrt(3) cm^2 ≈ 86.6 cm^2.
Volume of an Octahedron
An octahedron can be decomposed into two square pyramids with equilateral triangular faces. If the side lengths are all L, then using the Pythagorean Theorem, you can calculate that each square pyramid has a height of L/sqrt(2). Therefore, the volume of each pyramid is (L^3)/(3*sqrt(2)). Since there are two of them, we get the volume formula
V = 2*(L^3)/(3*sqrt(2))
V = [sqrt(2)/3]*L^3
Another way to conceptualize an octahedron besides the union of two square pyramids is to consider a regular tetrahedron of edge length 2L. Now imagine cutting off the four corners to make four smaller tetrahedra of edge length L. The figure left in the middle is a regular octahedron with an edge length of L. The volume of this shape is the volume of the larger tetrahedron minus the volumes of the four smaller tetrahedra. This produces the same volume equation as above. Both geometric decompositions are shown below.
Two Ways to See an Octahedron
A regular octahedron has a volume of 371.92 cubic centimeters. What is its surface area?
For this problem, we need to work backwards from the volume to find the edge length L, then plug L into the surface area formula. Doing this gives us
371.92 = [sqrt(2)/3]L^3
371.92/[sqrt(2)/3] = L^3
788.961462 = L^3
L = 9.24 cm
S = 2*sqrt(3)*L^2
S = 32.01 cm^2
An octahedron's surface area equals its volume. What is the edge length?
For this problem, we need to equate the formulas for V and S and solve for L. This gives us
[sqrt(2)/3]L^3 = 2*sqrt(3)L^2
L^3 / L^2 = 6*sqrt(3)/sqrt(2)
L = 3*sqrt(6)
L ≈ 7.3485
Suppose you want to make a closed octahedral box and the only materials you have are two sheets of paper in the shape of equilateral triangles with a side length of 9 inches. If you use all of the paper without any waste, what is the volume of the resulting box?
Solution: Each triangle can be cut into four triangles with a side length of 4.5 inches, for a total of eight equilateral triangular pieces. Since L = 4.5 inches, the volume is (sqrt(2)/3)4.5^3 42.9567 cubic inches. Below is a diagram of how to cut the triangles to form the octahedral faces.
Distances Inside the Octahedron
If an octahedron has an edge length of L, then the distance between opposite vertices is sqrt(2)*L. The distance between opposite (parallel) faces is [sqrt(6)/3]*L. These facts are used in the next example.
An octahedron is inscribed within a larger sphere, and a smaller sphere is inscribed within the octahedron. What is the ratio of the volumes of the two spheres?
If the octahedron has an edge length of L, then the radius of the outer sphere is one half the distance between opposite vertices of the octahedron, so R1 = L/sqrt(2). The radius of the smaller sphere is half the distance between opposite faces of the octahedron, so R2 = [sqrt(6)/6]L. Using the formula for the volume of a sphere, we have the volumes
Outer = [4π/3](L/sqrt(2))^3 = π(sqrt(2)/3)L^3
Inner = [4π/3](L*sqrt(6)/6)^3 = π(sqrt(6)/27)L^3
The ratio of the outer spheres volume the inner sphere's volume is (sqrt(2)/3) / (sqrt(6)/27) = 3*sqrt(3) ≈ 5.1962. Curiously, if you make the same calculation for a cube, you also get a ratio of 3*sqrt(3) for the outer and inner sphere volumes.
Other Facts About the Regular Octahedron
- The dihedral angle between any two adjacent faces is arccos(-1/3) ≈ 109.47°. This is the tetrahedral angle, the angle between four points spaced equally on a sphere, an angle often encountered in chemistry when looking at bonds between elements in a molecule such as methane, CH4.
- In nature, crystals of diamond and fluorite are octahedral.
- The polyhedral dual of an octahedron is the cube.
- The octahedron is used as an 8-sided die.
Octahedral numbers are those that represent the number of spheres packed in an octahedral arrangement, i.e., close-packing. The first ten octahedral numbers are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,... The sixth octahedral number, 146, is represented in the image. The nth octahedral number has an edge length of n balls and is given by the formula
Oct(n) = (2n^3 + n)/3
For example, the 100th octahedral number is (2*100^3 + 100)/3 = 666700.