Table of Contents

Last modified on August 3rd, 2023

A pentagonal prism is a three-dimensional solid consisting of two identical pentagonal bases connected by five lateral faces. It is a heptahedron. It has 7 faces, 15 edges, 10 vertices.

A 2-dimensional net can be used to construct a pentagonal prism to understand its shape. Generally a net is made from paper and cut in a manner such that it can be folded and modified into 3-D shape.

A pentagonal prism can be can also be **right** or **oblique**, depending on the alignment of its bases. It can also be **regular** or **irregular** based on the uniformity of its cross-section.

A right pentagonal prism is a prism where the lateral faces are perpendicular to the bases. Thus, the lateral faces are rectangular. Therefore, two bases appear above one another when the prism rests on its base.

A right pentagonal prism can be regular when the base is a regular pentagon with equal base edges as shown in the above diagram or irregular when its base is an irregular pentagon with unequal base edges.

An oblique pentagonal prism is a slanted prism where the lateral faces are not perpendicular to the bases. So the lateral faces are parallelogram-shaped. Therefore, the two pentagonal bases do not appear above one another when the prism rests on its base.

Like all other polyhedrons, we can calculate the surface area and volume of a pentagonal prism.

The formula is given below:

**Total Surface Area ( TSA) = 5ab + 5bh**, here

Also**, **since **Lateral Surface Area ( LSA)** = Base perimeter x height

Here, Base Perimeter = 5 x base edge x height = 5bh

∴**Lateral Surface Area ( LSA)** =

Thus, we can write

**Total Surface Area ( TSA) = 5ab + LSA**

**Find the lateral and the total surface area of a pentagonal prism with a base edge of 7 cm, an apothem of 4.81 cm and a height of 3 cm.**

Solution:

As we know,

Lateral Surface Area (*LSA) = 5bh,* here *b* = 7 cm, *h* = 3 cm

∴ *LSA* = 5 × 7 × 3

= 105 cm^{2}

Total Surface Area (*TSA*) = 5*ab* +* LSA, *here a = 4.81 cm, *LSA* = 105 cm^{2}

∴*TSA* = 5 × 4.81 × 7 + 105

= 273.35 cm^{2}

Finding the surface area of a pentagonal prism when the **BASE EDGE** and **HEIGHT** are known

**Find the surface area of a pentagonal prism with a base edge of 10.5 in and height 7 in.**

Solution:

Here we will use an alternative formula.

Total Surface Area (*TSA*) = ${ 5bh+\dfrac{1}{2}\sqrt{5\left( 5+2\sqrt{5}\right) }b^{2} }$ , here *b* = 10.5 cm, *h* = 7 in.

∴ *TSA* = ${ 5\times 10.5\times 7+\dfrac{1}{2}\sqrt{5\left( 5+2\sqrt{5}\right) }\times \left( 10.5\right) ^{2}}$

≈ 746.9 in^{2}

The formula is given below:

**Volume ( V) = **

Let us solve some examples to understand the concept better.

**Find the volume of a pentagonal prism with an apothem of 5.5 cm, a base edge of 8 cm and a height of 6 cm.**

Solution:

As we know,

Volume (*V*) = ${\dfrac{5}{2}abh}$*, *here *a* = 5.5 cm, *b* = 8 cm, *h* = 6 cm

∴*V* = ${\dfrac{5}{2}\times 5.5\times 8\times 6}$

= 660 cm^{3}

**More Resources:**- Volume of a Prism
- Surface Area of a Prism
- Right Prism
- Oblique Prism
- Rectangular Prism
- Volume of a Rectangular Prism
- Surface Area of a Rectangular Prism
- Triangular Prism
- Volume of a Triangular Prism
- Surface Area of a Triangular Prism
- Hexagonal Prism
- Volume of a Hexagonal Prism
- Surface Area of a Hexagonal Prism
- Pentagonal Prism
- Volume of a Pentagonal Prism
- Surface Area of a Pentagonal Prism
- Trapezoidal Prism
- Volume of a Trapezoidal Prism
- Surface Area of a Trapezoidal Prism
- Square Prism
- Volume of a Square Prism
- Surface Area of a Square Prism
- Octagonal Prism
- Heptagonal Prism
- Decagonal Prism

Last modified on August 3rd, 2023