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CAHSEE Math Exam: Help and Review21 chapters | 242 lessons

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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.

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.

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

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.

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.

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

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

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 2*x*^2 = *c*^2. Square root both sides and you have *c*=*x*(square root of 2).

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=4*bh*, 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 4*bh*. 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.

An octahedron consists of two square pyramids. To find the volume of an octahedron, find the volume of one the pyramids and multiply by two. The volume of a pyramid is *b*^2(*h*)/3, where *b* is the length of one side of the square that forms the base and *h* is the height of the pyramid. The formula for the volume of an octahedron would be 2*b*^2(*h*) / 3.

To find the volume of an octahedron, you need to know the length of the square base and the height of the pyramid. In this example, the length of the square base is 12 inches (the same as the base of the triangle) and the height of the pyramid is 8.5 inches. The formula for volume of an octahedron is *V*=2*b*^2(*h*)/3. Therefore the volume of the octahedron is 2(144)(8.5) / 3 = 816. Volume is measured in cubic units, so the volume of this octahedron is 816 cubic inches.

Let's review. **The octahedron** is one of five solids in the set of Platonic solids, and the one with eight faces. **The Platonic solids** are solid shapes that the philosopher Plato assigned each to be a representation of the natural elements. The octahedron is created by connecting two square pyramids. It has eight faces, six vertices, and twelve edges. The surface area can be obtained using the formula Surface Area = 4*b**h*. The formula for the volume is *V* = *b*^2(*h*) / 3.

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CAHSEE Math Exam: Help and Review21 chapters | 242 lessons

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