A pentagonal prism is a prism that has two pentagonal bases like top and bottom and five rectangular sides. It is a type of heptahedron with 7 faces, 10 vertices and 15 edges.

**Face**: A flat side of 3 dimensional object**Base**: One of two parallel and congruent sides of an object**Edge**: Intersection of two faces on a solid object. This is a line.**Vertex**: Joining point of two edge sides.

Prism is a three dimensional box that have two common bases. A pentagonal prism can have pentagonal bases which gives five sides. A pentagonal prism is also known as five- sided polygon prism. If all the sides of the of the pentagonal prism are of equal and same length, then it is said to be a regular pentagonal prism. The two important measures made on a pentagonal prism is to find its volume and surface area.

A prism is a right pentagonal prism when it has two congruent and parallel pentagonal faces and five rectangular faces that are perpendicular to the triangular ones. In regular pentagonal prism, all the rectangular faces are congruent. The rectangular faces are said to be lateral, when the pentagonal faces are bases. The lateral side faces are called lateral edges.

## Volume of Pentagonal Prism

To find the volume of regular pentagonal prism, first you have to find the apothem length (a). The apothem length is a measure from the centre of a polygon to the midpoint of any side. The formula to find the volume of pentagonal prism is given as

Volume of pentagonal prism= \(\frac{5}{2}abh\) cubic units.

Where,

a = Apothem length of the pentagonal prism

b = Base length of the pentagonal prism

h = Height of the pentagonal prism

## Surface Area of Pentagonal Prism

The surface area is the area that describes the material that will be used to cover a geometric solid shapes. The formula to find the surface area of the pentagonal prism is given by

Surface area of pentagonal prism = 5ab + 5bh square units

Where,

a = apothem length of the pentagonal prism

b = base length of the pentagonal prism

h = height of the pentagonal prism

## Solved problem

### Question:

Find the surface area and volume of the pentagonal prism with apothem length of 6 cm , base length of 10 cm and height of 11 cm ?

### Solution :

Given,

Apothem Length of the pentagonal prism, a = 6 cm

Base length of the pentagonal prism, b = 10 cm

Height of the pentagonal prism, h = 11 cm

Volume and the surface area of pentagonal prism

Volume of the pentagonal prism = \(\frac{5}{2}abh\) cu.units

= \(\frac{5}{2}(6\times 10\times 11)\)

= \(\frac{5}{2}(660)\)

= 5 x 330

= 1650

Therefore, the volume of pentagonal prism is 1650 cm^{3 }.

Surface area of pentagonal prism = 5ab + 5bh square units

= 5 (6×10) + 5 (10×11)

= 5(60) + 5(110)

= 300 + 550

= 850

Therefore, the surface area of pentagonal prism is 850 cm^{2 }.