Prism Formula

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. 

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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 is 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. 

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Different types of Prisms

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

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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)

2. The surface area of a triangular prism formula = (Apothem length x base length) + 3 x (base length x height)

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

4. 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 

Find the Volume of Prism

The volume of the prism may be calculated using the procedures below:

• First, write down the prism's dimensions.

• Using the formula V = BH, find the volume of the prism, where "V", "B", and "H" are the volume, base area, and height of the prism, respectively.

• After you've calculated the volume of the prism, write the unit of volume of the prism at the end (in terms of cubic units).

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.

Difference Between Pyramid and Prism

• Both a pyramid and a prism are three-dimensional polyhedron-shaped structures, with the difference in their bases being the most significant.

• A prism, on the other hand, has two bases, whereas a pyramid has just one.

• A pyramid's sides are always triangular, whereas the sides of a prism are always rectangular.

• The sides of a pyramid are angled to the base, whereas the sides of a prism are perpendicular to the base.

• A pyramid's sides are always connected together at a point; whereas, a prism's sides are not always joined together at a point.

• In a pyramid, the point where all of the sides meet is called the apex or vertex, and it is located vertically above the center of the base, but in a prism, there is no such point.

• A pyramid is associated with the field of geometry, whereas a prism is associated with the fields of geometry and optics.

• The existence of an apex distinguishes a pyramid. There is no apex in Prism.