A **pentagonal prism** It is a three-dimensional geometric figure whose identical bases are pentagon-shaped, and also has a total of 5 parallelogram-shaped faces.

If the faces are rectangular, it is said to be a *right pentagonal prism*while if the edges are inclined with respect to the bases, then it is a *oblique pentagonal prism*. In the following image there is an example of each one.

The base pentagon can be regular if its five sides have the same measure, as well as the internal angles, otherwise it is an irregular pentagon. If the base of the prism is regular, it is *regular pentagonal prism.* otherwise it is a prism *irregular pentagonal*.

The pentagonal prism is a harmonious structure used in architecture and object design, such as the modern building shown in the figure above. The windows in the shape of an irregular pentagon form the base of the prisms.

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**Characteristics of the pentagonal prism**

-It is a three-dimensional geometric figure, the surfaces that compose it enclose a certain volume.

-Their bases are pentagons and their lateral faces can be rectangles or parallelograms.

-It has vertices -the corners of the prism- and edges -edges or edges-.

-If the edges that join the bases are perpendicular to them, the prism is straight, and if they are inclined, the prism is oblique.

-When the base is a pentagon whose internal angles are less than 180º, the prism is *convex*but if one or more interior angles is greater than 180º, it is a prism *concave*.

**Elements of the pentagonal prism**

–**Bases**: It has two pentagonal and congruent bases -their measurements are the same-, either regular or irregular.

–**faces**: A pentagonal prism has a total of 7 faces: the two pentagonal bases and the five parallelograms that make up the sides.

–**Edge**: segment that joins two bases, shown in red in figure 3 or the one that joins two sides.

–**Height**: distance between the faces. If the prism is straight, this distance is equal to the size of the edge.

–**Vertex**: point in common between a base and two lateral faces.

The lower figure shows a right pentagonal prism with a regular base, in which the segments that form the base have the same measure, called *to*.

This type of prism also has the following elements, typical of the regular pentagon:

–**radius R**: distance between the center of the pentagon and one of the vertices.

–**Apothem LA**: segment that joins the center with the midpoint of one of the sides of the pentagon.

**How many vertices does a pentagonal prism have?**

In a pentagon there are 5 vertices and since the pentagonal prism has two pentagons as bases, this body has a total of 10 vertices.

**How many edges does a pentagonal prism have?**

The number of edges for geometric solids with flat faces, such as prisms, can be calculated using the *Euler’s theorem *for convex polyhedra. Leonhard Euler (1707-1783) is one of the greatest mathematicians and physicists in history.

The theorem establishes a relationship between the number of faces, which we will call C, the number of vertices V, and the total number of edges A as follows:

*C+V = A+2*

For the pentagonal prism we have: C = 7 and V = 10. Solving for A, the number of edges:

*A = C+V-2*

Substituting values:

A = 7 + 10 – 2 = 15

A pentagonal prism has 15 edges.

**How to get the volume of a pentagonal prism?**

The volume of the pentagonal prism measures the space enclosed by the sides and the bases. It is a positive quantity that is calculated by the following property:

*Any plane that cuts the prism perpendicular to its edges generates an intersection with the same shape as the base, that is, a pentagon with the same dimensions.*

Therefore, the volume of the pentagonal prism is the product of the area of the base and the height of the prism.

Be *AB* the area of the pentagonal base and *h* the height of the prism, then the volume *V* is:

*V = AB xh*

This formula is of a general nature, being valid for any prism, be it regular or irregular, straight or oblique.

The volume of a prism is always given in cubed units of length. If the length of the sides and the height of the prism are given in meters, then the volume is expressed in m3, which is read “cubic meters”. Other units include cm3, km3, inches3, and more.

**– Volume of regular pentagonal prism**

In a regular pentagonal prism the bases are regular pentagons, which means that the side and internal angles are equal. Given the symmetry of the body, the area of the pentagon and therefore the volume are easily calculated in several ways:

**Knowing the height and the measure of the side**

Be *to* the measure of the side of the pentagonal base. In that case the area is calculated by:

Therefore the volume of the regular pentagonal prism of height h is:

*V = 1.72048 a2⋅h*

**Knowing the height and the measure of the radius **

When you know the *radius R* of the pentagonal base, this other equation can be used for the area of the base:

*A = (5/2)R2⋅ sin 72º*

In this way the volume of the pentagonal prism is given by:

*V = (5/2)R2 ⋅ h ⋅ sin 72º*

* *Where *h* is the height of the prism

**Knowing the height, the measure of the apothem and the value of the perimeter**

The area of the pentagonal base can be calculated if its perimeter P is known, which is simply the sum of the sides, as well as the measure of the apothem LA:

*A = P. LA / 2*

Multiplying this expression by the value of the height *h*we have the volume of the prism:

V = *P.LA .h / 2*

**– Volume of the irregular pentagonal prism**

The formula given at the beginning is valid even when the base of the prism is an irregular pentagon:

*V = AB xh*

Various methods are used to calculate the area of the base, for example:

-Method of triangulation, which consists of dividing the pentagon into triangles and quadrilaterals, whose respective areas are easily calculated. The area of the pentagon will be the sum of the areas of these simpler figures.

-Method of Gauss’s determinants, for which it is necessary to know the vertices of the figure.

Once the value of the area is determined, it is multiplied by the height of the prism to obtain the volume.

**References**

Alexander, D. 2013. Geometry. 5th. Edition. Cengage Learning.

Math Open Reference. Area of a polygon. Retrieved from: mathopenref.com.

Universe Formulas. Euler’s theorem for polyhedra. Recovered from: universoformulas.com.

Universe Formulas. Area of a regular pentagon. Recovered from: universoformulas.com.

Wikipedia. Prism. Recovered from: es.wikipedia.com.

Wikipedia. pentagonal prism. Recovered from: es.wikipedia.com.