# Pentagonal Prism

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.

## Right Pentagonal Prism

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.

## Oblique Pentagonal prism

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.

## Formulas

### Surface Area

The formula is given below:

Total Surface Area (TSA) = 5ab + 5bh, here a = apothem, b = base edge, h = height, lateral surface area

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

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

Lateral Surface Area (LSA) = 5bh

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 cm2
Total Surface Area (TSA) = 5ab + LSA, here a = 4.81 cm, LSA = 105 cm2
TSA = 5 × 4.81 × 7 + 105
= 273.35 cm2

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 in2

### Volume

The formula is given below:

Volume (V) = ${\dfrac{5}{2}abh}$, here a = apothem, b = base edge, h = height

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 cm3

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