Surface Area and Volume Help

Introduction to Polyhedrons

In Euclidean three-space, geometric solids with straight edges have flat faces, also called facets, each of which forms a plane polygon. An object of this sort is known as a polyhedron .

The Tetrahedron

A polyhedron in 3D must have at least four faces. A four-faced polyhedron is called a tetrahedron. Each of the four faces of a tetrahedron is a triangle. There are four vertices. Any four specific points, not all in a single plane, form a unique tetrahedron.

Surface Area Of Tetrahedron

Figure 8-1 shows a tetrahedron. The surface area is found by adding up the interior areas of all four triangular faces. In the case of a regular tetrahedron , all six edges have the same length, and therefore all four faces are equilateral triangles. If the length of each edge of a regular tetrahedron is equal to s units, then the surface area, B , of the whole four-faced regular tetrahedron is given by:

B = 3 ^1/2 s ²

where 3 ^1/2 represents the square root of 3, or approximately 1.732. This also happens to be twice the sine of 60°, which is the angle between any two edges of the figure.

Fig. 8-1 . A tetrahedron has four faces (including the base) and six edges.

Volume Of Tetrahedron

Imagine a tetrahedron whose base is a triangle with area A , and whose height is h as shown in Fig. 8-1. The volume, V , of the figure is given by:

V = Ah /3

Surface Area and Volume of a Pyramid

The Pyramid

Figure 8-2 illustrates a pyramid . This figure has a square or rectangular base and four slanted faces. If the base is a square and the apex (the top of the pyramid) lies directly above a point at the center of the base, then the figure is a regular pyramid , and all of the slanted faces are isosceles triangles.

Fig. 8-2 . A pyramid has five faces (including the base) and eight edges.

Surface Area Of Pyramid

The surface area of a pyramid is found by adding up the areas of all five of its faces (the four slanted faces plus the base). In the case of a regular pyramid where the length of each slanted edge, called the slant height , is equal to s units and the length of each edge of the base is equal to t units, the surface area, B , is given by:

B = t ² + 2 t ( s ² − t ² /4) ^1/2

In the case of an irregular pyramid , the problem of finding the surface area is more complicated, because it involves individually calculating the area of the base and each slanted face, and then adding all the areas up.

Volume Of Pyramid

Imagine a pyramid whose base is a square with area A , and whose height is h as shown in Fig. 8-2. The volume, V , of the pyramid is given by:

V = Ah /3

This holds true whether the pyramid is regular or irregular.

Surface Area and Volume of a Cube

The Cube

Figure 8-3 illustrates a cube . This is a regular hexahedron (six-sided polyhedron). It has 12 edges, each of which is of the same length. Each of the six faces is a square.

Fig. 8-3 . A cube has six square faces and 12 edges of identical length.

Surface Area Of Cube

Imagine a cube whose edges each have length s , as shown in Fig. 8-3. The surface area, A , of the cube is given by:

A = 6 s ²