In a prism there are two equal and similar end faces, parallel to each other, joined together by other rectangular faces. In some cases, the faces joining the end faces may be parallelograms instead of rectangles. There are many types of Prism, and they are named after the shape of their base. A prism having pentagonal base faces is called a pentagonal prism. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.

## What is a Pentagonal Prism

The prisms are named after the base. If the base is a triangle, the prism is called right triangular prism. If the base is a pentagon, the prism is right pentagonal prism. A prism having pentagonal end faces is called a pentagonal prism.

### Regular Pentagonal Prism

A right regular prism is the right angled prism whose all lateral faces are equal rectangles. The end faces of the regular prism are regular polygons. Similarly, a regular pentagonal is the right angled prism whose all lateral faces are equal rectangles and end faces of the regular pentagonal prism are regular pentagons.

### Oblique Pentagonal Prism

A oblique prism is a prism in which the parallel lateral edges are oblique to the base edges at their points of intersection. Oblique pentagonal prism, the prism whose axis is inclined to the end faces called an oblique prism.

## Volume of a Pentagonal Prism

The pentagonal prism is a prism having two pentagonal bases and five rectangular sides. The volume for prism is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base.

### Volume of a Pentagonal Prism Formula

The formula for the volume of any right prism,

=> V = Ah

Where, A = area of the base and h = perpendicular height

Now

In Pentagonal Prism:

Area of pentagonal base (A) = $\frac{5}{2}$ab

=> Volume of Prism = Ah = $\frac{5}{2}$abh

Where, a = apothem length, b = side, h = height.

Formula:

Volume of a Pentagonal Prism (V) = Ah = $\frac{5}{2}$abh

Where, a = apothem length, b = side, h = height.

## Surface Area of a Pentagonal Prism

The surface area of a polyhedren is the total of the area of the polygons that form its faces. For prisms, surface area is the sum of area of two bases and several parallelograms. The surface area of the prism is equal to the perimeter of the base times the sum of the apothem of the polygon and the height of the prism.

### Surface Area of a Pentagonal Prism Formula

Surface area of any prism is given by,

Surface Area of a Prism = L + 2B

Where, L is the lateral surface area and B is the base area of the prism.

Now, surface area of the pentagonal prism

=> S = 2($\frac{1}{2}$ a P) + h P

= a P + h P

= P(a + h)

=> S = P(a + h)

Where, a = apothem height of the base and h = height of the prism.

Formula:

Surface Area of a Pentagonal Prism = P(a + h)

Where, a = apothem height of the base and h = height of the prism.

Example:

In right pentagonal prism, the apothem of the regular pentagon is 6 cm, the side of the pentagon are 18 cm each, and the height of the prism is 14 cm. Find the surface area of the prism.

Solution:
Given:

Apothem height (a) = 6 cm
Side of prism = 18 cm
Height of the prism (h) = 14 cm

Step 1:

The surface area is equal to the perimeter of the base times the sum of the apothem of the polygon and the height of the prism.

Since each side of pentagonal prism is 18 cm

=> Perimeter of the base = 5 x 18 = 90 cm

Step 2:

Surface Area of a Pentagonal Prism (S) = P(a + h)

=> S = 90(6 + 14)

= 90 x 20

= 1800

The surface area of the pentagonal prism is 1800 cm2 .