Surface Area of Prisms (Definition, Basics, Branches, Facts, Examples) - BYJUS

# Surface Area of Prisms

A prism is a three–dimensional shape whose ends are similar, and sides are made up of parallelograms. We will learn the different types of prisms, the concept of nets of prisms, and the steps involved in calculating the surface area of a prism....Read MoreRead Less

## What is a Prism?

A prism is a three-dimensional object or shape with two identical surfaces that face one another. The bases of a prism refer to each of these faces. A polygon such as a triangle, rectangle, square, or even a pentagon can be used as the base of a prism. The remaining faces of a prism can be either a parallelogram or a rectangular shape.

## What do we understand by Surface Area?

The surface area (S) of a prism is equal to the sum of the areas of the prism’s bases and lateral faces. It is measured in square units like square inches, square meters, and square feet. There are two bases and four lateral surfaces in a prism.

The formula for the surface area of a prism is the summation of the areas of the bases and the lateral faces.

S = areas of bases + areas of lateral faces

## What is a Net?

A shape’s “net” (also known as a geometry net) is a term that describes how a 3D shape would look if it were opened out and laid flat. When a three-dimensional shape is unfolded, it looks like a net. To make 3D shapes, you can fold the nets. Multiple nets can be used to create a 3D shape.

## What is a Rectangular Prism?

A rectangular prism is a three-dimensional shape that has all its faces rectangular in shape.

A rectangular prism is also known as a cuboid because of its shape. Examples of a rectangular prism are geometry boxes, notebooks, diaries, rooms, and other places. The shape of a rectangular prism can be seen in the diagram below:

## Properties of a Rectangular Prism

• A rectangular prism has six faces, eight vertices, and twelve edges.

• There are three dimensions to it: length, width, and height.

• The opposite faces of a rectangular prism are congruent.

## The Formula for the Surface Area of a Rectangular Prism

A rectangular prism has six faces. Any 2 of its opposite faces can act as bases and the other 4 as lateral faces. For a rectangular prism, the sum of the areas of the faces can be calculated as follows:

S = 2lw + 2lh + 2wh

Where, l, w, and h are the length, width, and height of the rectangular prism, respectively.

## What is a Cube?

A cube is a three-dimensional solid object with six square faces. The cube is one of the five Platonic solids and is the only regular hexahedron. It is made up of six faces, twelve edges, and eight vertices.

Example: Find the surface area of the rectangular prism.

Solution:

We have to find the area of each face by using a net:

Top: $$5\times 3=15$$                  Area = $$l\times b$$

Bottom: $$5\times 3=15$$            Area = $$l\times b$$

Side: $$5\times 4=20$$                Area = $$l\times b$$

Side: $$5\times 4=20$$                Area = $$l\times b$$

Front: $$3\times 4=12$$               Area = $$l\times b$$

Back: $$3\times 4=12$$               Area = $$l\times b$$

The sum of all the faces will be the surface area of the prism.

Surface Area = Area of top + Area of bottom + Area of side + Area of side + Area of front + Area of back

S = 15 + 15 + 20 + 20 + 12 + 12 = 94 cm²

Hence, the surface area of the prism is 94 cm².

## What is a Triangular Prism?

A polyhedron with two triangular bases and three rectangular lateral faces is known as a triangular prism. We have nine distinct nets in the triangular prism.

## Properties of a Triangular Prism

The properties of a triangular prism are listed below:

• It has a total of nine edges, five faces, and six vertices.

• It has three rectangular faces and two triangular bases.

• The triangular faces are congruent.

## The Formula for the Surface Area of a Triangular Prism

A triangular prism has five faces. There are two triangular opposite faces called the bases, and three of its lateral faces are rectangular. For a triangular prism, the sum of the areas of the two bases and the three lateral faces can be calculated as:

S = L × a + b + c + 2 × base area

S = L × a + b + c + b × h

Example: Find the surface area of a triangular prism.

Solution: We have to find the area of each face by using a net:

Bottom: $$8\times 4=32$$                  Area = $$l\times b$$

Front: $$\frac{1}{2}\times 8\times 6=24$$              Area = $$\frac{1}{2}\times b\times h$$

Back: $$\frac{1}{2}\times 8\times 6=24$$              Area = $$\frac{1}{2}\times b\times h$$

Side: $$9\times 4=36$$                      Area = $$l\times b$$

Side: $$4\times 6=24$$                      Area = $$l\times b$$

The sum of all the faces will be the surface area of the prism.

Surface Area = Area of bottom + Area of side + Area of side + Area of front + Area of back

S = 32 + 24 + 24 + 36 + 24 = 140

Hence, the surface area of the prism is 140 m².