# Prism Formula

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## What Is A Prism?

The question of what is a prism has a very simple answer. A prism is a five-sided polyhedron with a triangular cross-section. In a prism, there are two identical parallel triangles opposite to each other. Along with the triangles, three rectangular surfaces are inclined to each other. A prism is a transparent solid used for refraction. The two inclined rectangular surfaces through which the light passes are called the refracting surfaces. The angle formed between these two refracting surfaces is called the refracting edge of the prism. The section of the prism that is perpendicular to the refracting edge is called the principal section of the prism. The third rectangular surface at the bottom is the base of the prism.

Again, the question of what is a prism can be answered in two ways as the concept of it is used in both mathematics as well as science. In mathematics, a prism defined as a polyhedron.  In physics (optics), a prism is defined as the transparent optical element that has flat and polished surfaces used for refracting light. There are two formulas of the prism.

1. The surface area of a prism = (2×BaseArea) +Lateral Surface Area

2. The volume of a prism =Base Area× Height

### Properties of Prism

Now that we know what is a prism, we can know the properties of prism easily.

1. Among all the properties of the prism the most basic is that the base and top of the prism are parallel and congruent.

2. In a prism, except the base and the top, each face is a parallelogram. These faces are known as Lateral face.

3. The base and the top has one edge common with every lateral face.

4. The height of the prism is basically the common edge of two adjacent side faces.

### Types of Prism

There are different types of prisms. Some of them are:

1. Rectangular Prism: In a Rectangular Prism, 2 rectangular bases are parallel to each other and 4 rectangular faces.

2. Triangular Prism: In a Triangular Prism, there are 2 parallel triangular surfaces, 2 rectangular surfaces that are inclined to each other and 1 rectangular base.

3. Pentagonal Prism: In a Pentagonal Prism, 2 pentagonal surfaces are parallel to each other and 5 rectangular surfaces that are inclined to each other.

4. Hexagonal Prism: In a Hexagonal Prism, there are 2 hexagonal surfaces parallel to each other and 6 rectangular surfaces that are inclined to each other.

These were the few different types of prisms.

Different types of Prisms

There can be yet two other types of prisms that can also be a right prism and oblique prism.

### Base Area

1. The base area of a rectangular prism formula = base length x base width.

2. The base area of a triangular prism formula = ½ x apothem length x base length.

3. The base area of a pentagonal prism formula = 5/2 x apothem length x base length.

4. The base area of a hexagonal prism formula = 3 x apothem length x base length.

### Surface Area Of A Prism

1. The surface area of a rectangular prism formula = 2 x (base length x base width) +

(base width x height) + (height x base length)

1. The surface area of a triangular prism formula = (Apothem length x base length) +

3 x (base length x height)

1. The surface area of a pentagonal prism formula = 5 x(apothem length x base length) +

5 x (base length x height)

1. The surface area of a hexagonal prismformula = 6 x (apothem length x base length) + 6 x (base length x height)

### Volume of A Prism

1. The volume of a rectangular prism formula = Base width x base length x height

2. The volume of a triangular  formula= ½ x apothem length x base length x height

3. The volume of a pentagonal prism formula = 5/2 x apothem length x base length x height

4. The volume of a hexagonal prism formula = 3 x apothem length x base length x height

### Solved Examples

Example 1) The base of a right triangle prism where the lengths of the sides arc are 13 cm, 20 cm, and 21 cm. If the height of the prism is 9 cm. Find:

1. The area of the total lateral surface.

2. Area of the whole surface.

3. The volume of the prism.

Solution 1) Let the semi-perimeter of the triangular base of the prism be s.

Therefore, S = (13 + 20 + 21)/2 cm. = 27 cm

The area of the prism = √[s(s - a)(s - b)(s - c)]

= √(27(27 - 13)(27 - 20)(27 - 21)) sq. cm.

= √(27 × 14 × 7 × 6) sq. cm.

= 9 × 7 × 2 sq. cm.

The area of the total lateral surface of the prism = (perimeter of the base) × height

= (486 + 2 × 126) sq. cm.

The volume of the prism = area of the base × height

= 126 × 9 cu.cm.

= 1134 cu.cm.

Example 2) Find the surface area of the triangular prism if the apothem length = 5 cm, base length = 10 cm, and height = 18 cm.

Solution 2) We have:

Apothem length = 5 cm;

Base length = 10 cm;

height = 18 cm

The surface area of a triangular prism = ab + 3bh

= (5 cm × 10 cm) + (3 × 10 cm × 18 cm)

= 50 cm2 + 540 cm2

= 590 cm2

### Fun Facts

1. The prism helps in the refraction of light. It splits the light into a lot of different colours which is known as a spectrum.

Question 1) How is light refracted through a Prism?

Answer 1) Light travels slowly in a glass than in air, so when a ray of light travelling in the one medium i.e., air enters into another medium, i.e., glass, the speed of light changes, and it moves slowly. It also suffers refraction. When the ray of light passes from the glass back to the air, it again faces refraction and the speed of it increases.

With each refraction suffered by the prism, the ray of light always bends towards the base of the prism.