How To Calculate Volumes Of Pentagonal Prisms

A prism can be an elegant decorative item, a tool in physics or merely an alluring geometric construct that also happens to be useful. The human eye and mind have a yen for symmetry in art and in nature, and they find attractiveness in three-dimensional shapes that are regular, multi-faceted and transmit as well as reflect light.

Objects with a lot of sides – for example, a dodecahedron, which has 12 identical five-sided faces making up its surface – are fun to look at, but the math underlying their geometry can be tedious at best.

A five-sided (that is, pentagonal) prism is a useful starting point for students trying to learn how to calculate the volumes of regular polyhedrons, of which prisms are one of many common types and an infinite number of theoretical types.

"Polyhedra" perhaps sounds like a monster from the world of Greek mythology. In fact, the "Greek" part of that is correct: The word polyhedra (singular polyhedron) means "many bases," and in the world of math, there is a lot you can do with those bases given their dimensions and angles.

A polyhedron is any three-dimensional solid consisting of plane faces. The face on which a polyhedron is depicted "resting" is its base, which can be identical to all, some or none of the other faces. The simplest example is a pyramid, which has four triangular faces. A cube has six identical faces and is a special case of a cuboid, which is any six-sided figure consisting of right angles.

A prism is a polyhedron that could have been created by "pushing" a polygon, or two-dimensional figure with three or more angles, in a straight line through space to form two ends and connecting them using as many parallel planes as the prism has sides. The simplest prism consists of two equilateral triangles with their faces parallel to each other and separated by three identical rectangular faces oriented at 60-degree angles to their neighboring faces.

A pentagonal prism the same thing expanded to include two additional angles and two more faces. It thus includes two pentagonal bases and five rectangular sides. It is therefore a heptahedron, because it has seven sides (hepta- is a Grrek prefix meaning "seven").

The area of any regular polygon (that is, one in which all angles and sides are identical) with side length s can be found from the formula:

A = (n)(s²)/[4 tan (180/n)]

For a pentagon (n = 5), this reduces to:

A = 5s²/2.91 = 1.72s²

If you were to "unfold" or "flatten" a pentagonal prism made of cardboard, you would be left with two identical pentagon faces (the bases of the prism) and five identical rectangular faces.

Two sides of each rectangle are shared with sides of the pentagons; call this length s. If you call label the other two sides (which can be as short or as long as you like, at least in theory) h, then the area of each rectangular side is sh, and the area of all of the sides combined is 5sh.

There are two pentagonal faces, so the total area of a pentagonal prism is:

A = 5(sh) + 2(1.72s²) = 5(sh) + 3.44s²

For any standard prism, the volume is just the area of the base times the height. That means multiplying 1.72s², the value for the area of a pentagon from the previous equation, by the height h in whatever units you are using. The volume formula is:

V = 1.72s²h

For example, if you have a large pentagonal prism with a height of 30 cm (0.3 m) and sides of 10 cm (0.1 m), the area is:

A = 5(sh) + 2(1.72s²) = 5(0.3 m)(0.1 m) + 2(1.72)(0.1 m)²

= 0.15 + 0.0344 = 0.1844 m²

The volume is given by:

V = (1.72)(0.1 m)²(0.3 m) = 0.00516 = 5.16 × 10^-3 m³