# Prism

A prism is a three-dimensional soild object having two identical and parallel shapes facing each other. The identical shapes are called the bases. The bases can have any shape of a polygon such as triangles, square, rectangle, or a pentagon. The diagram below shows a triangular prism.

## Definition

A prism is a member of the polyhedron family consisting of two identical and parallel polygonal bases. The bases are connected by flat faces forming a uniform cross-section.

In general, a prism refers to a transparent solid used to refract or scatter a beam of white light. It is a commonly used instrument in physics.

## Parts

A prism has bases, lateral faces, edges, and vertices.

• Base – The parallel faces which makes the 2 ends of any prism. They are congruent. The base determines the cross-section of any prism and it remains uniform throughout the shape.
• Lateral faces – The non-parallel faces which connects the 2 bases.
• Vertices – The corners.
• Edges – Where any 2 faces meet.

## Shapes

Depending on the base, a prism can be of different shapes. Some common shapes are: triangular, rectangular, square, pentagonal, hexagonal, heptagonal, octagonal, and trapezoidal. For example, a triangular prism has a triangular base and a square prism has a square base, here are some more shapes:

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

## Regular and Irregular Prisms

The diagram shows the difference between a regular and irregular triangular prism.

1. Regular Prism – It has a base which is a regular polygon with equal side lengths. Regular prisms have identical bases and identical lateral faces. Thus, all the above examples of prisms are regular such as triangular, rectangular, and pentagonal.
2. Irregular Prism – It has a base which is an irregular polygon with unequal side lengths. Irregular prisms have identical bases. However, the lateral faces are not identical.

## Right and Oblique Prisms

The diagram shows the difference between a right and an oblique pentagonal prism.

1. Right PrismIts lateral faces are perpendicular to its bases. The 2 bases of a right prism is aligned perfectly over one another.
2. Oblique Prism – It is a slanted prism. So its lateral faces are not perpendicular to its bases. The 2 bases are not aligned perfectly over one another.

Like all other polyhedrons, a prism also has a surface area and a volume.

## Formulas

### Surface Area

Lateral Surface Area (LSA) = Perimeter × Height

Total Surface Area (TSA) = (2 × Base Area) + LSA

TSA= (2 × Base Area) + (Perimeter × Height)

here, height is the distance between the 2 bases or the length of the prism.

### Volume

Volume = Base Area × Height, here, height is the distance between the 2 bases or the length of the prism.

Let us solve some examples involving prisms and the above formulas

## Solved Examples

Find the surface area of a triangular prism given whose base area is 12 cm2, perimeter is 16 cm, and length is 7 cm.

Solution:

As we know,
Total Surface Area (TSA) = (2 × Base Area) + (Perimeter × Height), here base Area = 12 cm2, perimeter = 16 cm, height = length = 7 cm
∴ TSA = (2 × 12) + (16 × 7)
= 136 in2

Find the total and lateral surface area of a rectangular prism whose base area is 36 cm2 , perimeter is 30 cm, and height is 11 cm.

Solution:

As we know,
Total Surface Area (TSA) = (2 × Base Area) + (Perimeter × Height), here base area = 36 cm2, perimeter = 30 cm, height = 11 cm
∴ TSA = (2 × 36) + (30 × 11)
= 402 cm2
Lateral Surface Area (LSA) = Perimeter × Height
∴ LSA = 30 × 11
= 330 cm2

Find the volume of a triangular prism whose base area is 64 cm2 and height is 7 cm.

Solution:

As we know,
Volume (V) = Base Area × Height
∴ V= B × h, here B = 64 cm2, h = 7 cm
= 64 × 7
= 448 cm3

## Real-life Examples

• Ice cubes and candy bars
• Books and notebooks
• A Rubik’s cube
• Camping tents and barns
• Glass or plastic prisms used in science laboratories to see spectrum of white light
• More Resources: