# Octahedron: Definition & Properties

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• 0:04 Definition of an Octahedron
• 0:33 Parts of an Octahedron
• 1:33 Distance from Vertex to Vertex
• 2:33 Finding the Surface Area
• 3:22 Finding the Volume
• 4:21 Lesson Summary

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Lesson Transcript
Instructor: David Karsner
In geometry, the octahedron is one of five Platonic solids. It is formed from eight equilateral triangles. If you took two congruent square pyramids and connected their bases, you would create an octahedron.

## Definition of an Octahedron

The octahedron is one of five solids in the set of Platonic solids, and the one with eight faces. Platonic solids get their name from Plato, not because he discovered them but because he mentioned them often in his writings. He assigned each solid to be a representation of the natural elements. The octahedron was the symbol for air. The other four Platonic solids are the tetrahedron, cube, dodecahedron, and the icosahedron.

## Parts of an Octahedron

The octahedron is a three-dimensional object and is composed of one- and two-dimensional parts. These parts have special names.

#### Base

The base of an octahedron is a square. If you picture an octahedron as two congruent square pyramids that have their bottoms touching, then the base of the octahedron is the square between the two pyramids.

#### Face

An octahedron has eight faces, which are all in the shape of equilateral triangles. These eight faces are where the solid gets its name. 'Octa' means eight. These faces form the surface area of the octahedron. The square that is the base of the octahedron is not part of the surface area; therefore, the base isn't also a face.

#### Edge

When two faces touch, the line segment that is formed is called an edge. An octahedron has 12 edges.

#### Vertex

When two edges intersect they form a vertex (plural being vertices). The octahedron has six vertices. Each vertex is formed when four edges intersect.

## Distance from Vertex to Vertex

Notice how each vertex has four edges that touch it. Those edges connect the vertex with four of the other five vertices. There is one vertex called the non-adjacent vertex that isn't connected to the vertex by an edge. The distance from any vertex to its non-adjacent vertex will always be the length of any edge times the square root of two (1.414). The line formed by creating the vertex to non-adjacent vertex is the hypotenuse of a right triangle. Two of the edges form the legs of the right triangle.

Since a right triangle has been created, you can use the Pythagorean Theorem a^2 + b^2 = c^2 where a and b are legs and c is the hypotenuse to find the distance between any vertex and its non-adjacent vertex. The edge is x in length. x^2 + x^2 = c^2, which simplifies to 2x^2 = c^2. Square root both sides and you have c=x(square root of 2).

## Finding the Surface Area

Since the surface of an octahedron consists of eight congruent triangles, the surface area of an octahedron would be the area of one of the triangles times eight. The formula is Surface Area=4bh, where b is the length of the base of the triangle and h is the length of the height of the triangle.

To find the surface area of an octahedron, you'll need to know the length of the base of the triangle and the height.

In this octahedron, the base is 12 inches and the height is 10.4 inches. The formula for surface area is 4bh. If we substitute 12 for b and 10.4 for h we have S.A. = 4(12)(10.4) = 499.2. Area is always measured in square units, so the surface area of this octahedron is 499.2 square inches.

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