**Introduction**

The boundary or outline of an object is called its shape. We come across many shapes in our daily lives and kids start recognising these shapes even before actually studying about them. The alphabets of English shapes are all shapes of different types. The Sun, the earth and other planets, the mountains and all other things in the world are all of the specific shapes. One such type of shape is the 3 – dimensional shape. 3 – dimensional shapes or 3D shapes are the shapes that have all the three dimensions, i.e. length, breadth and height. What is the surface area of 3-dimensional shapes?

**What is surface area?**

The surface area of any given object is the area or region occupied by the surface of the object. The surface area of 3 D shapes is similar to the area of 2 D shapes. Recall that the space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area. It is also measured in square units. Each 3 – dimensional shape has its own formula for finding its surface area. The surface area of the 3 – dimensional shapes can also be calculated using nets.

**What are nets?**

A net is a two-dimensional shape that can be folded to make a three-dimensional shape. In other words, the net of a solid is a diagram drawn on paper which when cut and folded along the lines can be used to construct a solid shape. Nets can be formed for any 3 – dimensional shape. For instance, below is the net diagram of a cylinder –

**How can nets be used to determine surface area?**

We know now that a net is a pattern made when the surface of a three-dimensional figure is laid out flat showing each face of the figure. How can we use the nets to find the surface area of a 3 – dimensional object? Net allows us to see all the 2-dimensional objects that make a 3 – dimensional object. The following steps are used to find the surface area of a 3 – dimensional object using nets –

- The first step is to decompose the 3 – dimensional object into rectangles, triangles, circles, squares and label each of the 2-dimensional shapes thus obtained.
- The next step is to find the dimensions which are the lengths of the sides that would be required to find the surface area.
- Now that we have the dimensions of each side, we can find out the area of each 2-dimensional shape that we had obtained in the first step.
- After fining the area of all the 2-dimensional shapes, the last step is to add up all the area to get the total area which is the surface area of the given 3 – dimensional shape.

The following points should be kept in mind while expanding a 3-dimensional shape in the form of nets –

- Make sure that the solid and the net have the same number of faces and that the shapes of the faces of the solid match the shapes of the corresponding faces in the net.
- Visualize how the net is to be folded to form the solid and make sure that all the sides fit together properly.

Let us now use the above steps to find the surface area of some of the common 3 – dimensional shapes using nets.

**Surface area of a cube using nets**

Cube is a common geometrical shape that we make use of in everyday lives. What is a cube? Let us find out.

**What is a cube?**

A cuboid whose length, breadth and height are equal is called a cube. Examples of a cube are sugar cubes, cheese cubes and ice cubes. In other words, it is an extension of a square in a 3D plane.

Below we have a general diagram of a cube

**Faces**

A cube has 6 rectangular faces, out of which all are identical.

**Edges**

A cube has 12 edges

**Vertx**

A cube has 8 vertices

**How to find the surface area of a cube?**

We know that a cube has six faces; therefore, the total surface area of the cube will be equal to the sum of all six faces of the cube. Also, we can observe from the shape of a cube that the surface of the cube is in a square shape. Therefore, the area of each face of the cube is equal to square of its edge. Let the length of the edge of the cube be “ c “. We will then have,

Using the formula of finding the area of a square, the area of one face = c ^{2}

Now, there are 6 faces in total, all of which have equal areas.

Therefore,

Total Surface area of a cube = c ^{2} + c ^{2} + c ^{2} + c ^{2} + c ^{2} + c ^{2} = 6 c ^{2}

So, the total surface area of a cube = 6 c ^{2}

Let us now see how to find the surface area of a cube using nets.

**Using nets to find the surface area of a cube **

Before, discussing the area of a cube using nets, let us observe how many nets for a cube can be possible? There can be 11 different possible nets of a cube. Below are the various nets possible –

This is important to understand so as to have clarity on the 3 – dimensional shape that has been unfolded so as to form a structure of nets.

Let us consider an example.

**Example**

Find the surface area of the following cube using nets –

**Solution**

We can see that each side of a cube has been given as 6 units. Let us first find the total surface area of the cube using the formula we had discussed above.

We know that the total surface area of a cube = 6 c ^{2} , where “ c “ is the side of the cube. Therefore,

The total surface area of a cube = 6 ( 6 ) ^{2} = 6 x 6 x 6 = 216 sq. units.

Let us now find out the surface area of the cube using nets.

The first step would be to unfold the given cube in the form of a net. We would obtain the following on doing so –

It can be clearly seen that the net of a cube is made from 6 equally-sized squares. Also, the area of a square is given by side ^{2}. Therefore, we will have

Area of one square of the net = 6 x 6 = 36 sq. units

There are 6 squares in the net, so, the area of the 6 squares will be 6 x 36 = 216 sq. units.

Hence, the total surface area of the given cube will be 216 sq. units.

We can see that the result obtained by using the formula and

the one obtained using the nets is the same.

**Surface area of a rectangular prism using nets**

**What is a rectangular prism?**

A rectangular prism is a polyhedron with two congruent and parallel bases. Some of the real-life examples of a rectangular prism are rooms, notebooks, geometry boxes etc. Following is the general representation of a rectangular prism.

Let us now discuss about the faces, vertices and edges of a rectangular prism. Before, that let us recall what we mean by faces, vertices and edges.

How many faces does a rectangular prism have? **It has 6 faces. **

How many edges does a rectangular prism have? **It has 12 vertices **

How many vertices does a rectangular prism have? **It has 8 vertices **

The following figure shows the representation of faces, vertices and edges in a rectangular prism-

**How to find the surface area of a rectangular prism?**

The total surface area of a rectangular prism is the sum of the areas of all of its faces. So, how do we calculate the total surface area of a rectangular prism?

The total surface area of a rectangular prism is the sum of the lateral surface area (LSA) and twice the base area of the rectangular prism. This means that –

**The total surface area of a rectangular prism = Lateral Surface Area + 2 x Area of the Base**

Therefore,

Total surface area of a rectangular prism = 2 ( l + b ) x h + 2 x Area of the Base

Now, area of the base = length x breadth = l x b

So,

Total surface area of a rectangular prism = 2 ( l + b ) x h + 2 x ( l x b )

⇒ Total surface area of a rectangular prism = 2 l h + 2 b h + 2 l b

⇒ Total surface area of a rectangular prism = 2 ( l h + b h + l b )

Hence,

**Total surface area of a rectangular prism = 2 ( l h + b h + l b )**

**Using nets to find the surface area of a rectangular prism**

What is the net of a polyhedron? We know that the net of any geometrical figure is obtained when it is unfolded along its edges and its faces are laid out in a pattern in two dimensions. By this definition, the net of a polyhedron is like an unfolded cardboard construction of that polyhedron. So, what will the net of a rectangular prism look like? The following is the net obtained when we open a rectangular prism along its surfaces. We can see that the nets of rectangular prisms are made up of rectangular and square shapes.

Let us consider an example.

**Example**

Find the surface area of the following rectangular prism using nets –

**Solution**

We can see that the length, breadth and height of the rectangular prism have been given as 8, 2 and 4 units respectively. Let us first find the total surface area of the rectangular prism using the formula we had discussed above.

We know that the Total surface area of a rectangular prism = 2 ( l h + b h + l b )

Therefore,

Total surface area of a rectangular prism = 2 ( 8 x 4 + 2 x 4 + 8 x 2 ) = 2 x (32 + 8 + 16 ) = 2 x 56 = 112 sq. units.

Let us now find out the surface area of the rectangular prism using nets.

The first step would be to unfold the given rectangular prism in the form of a net. We would obtain the following on doing so –

It can be clearly seen that the net of a cube is made from 6 differently sized rectangles. Also, the area of a rectangle is given by length x breadth. Let us label each rectangle. We will have,

Now,

Area of rectangle 1 = 8 x 2 = 16 sq. units

Area of rectangle 2 = 4 x 2 = 8 sq. units

Area of rectangle 3 = 8 x 4 = 32 sq. units

Area of rectangle 4 = 4 x 2 = 8 sq. units

Area of rectangle 5 = 8 x 2 = 16 sq. units

Area of rectangle 6 = 8 x 4 = 32 sq. units

Hence, the total surface area of the given rectangular prism will be 16 + 8 + 32 + 8 + 16 + 32 = 112 sq. units.

We can see that the result obtained by using the formula and the one obtained using the nets is the same.

**Surface area of a triangular prism using nets**

We know that a prism is a solid whose side faces are parallelograms and whose ends (bases) are congruent parallel rectilinear figures. A prism is a polyhedron that has two congruent and parallel polygons as bases. The rest of the faces are rectangles. One of such prisms is the triangular prism.

**What is a triangular prism?**

A solid whose lateral faces are rectangular and the bases are congruent triangles is called a triangular prism.

**Faces**

A triangular prism 3 rectangular faces and 2 triangular bases.

**Edges**

A triangular prism has 9 edges.

**Vertx**

A triangular prism has 6 vertices.

Below is the general figure of a triangular prism –

**Right Triangular Prism** – A right prism is called a right triangular prism if its ends are triangles. In other words, a triangular prism is called a right triangular prism if its lateral edges are perpendicular to its ends.

**How to find the surface area of a triangular prism?**

The surface area of a triangular prism is given by ab + 3bh where, “ a “ is the side, “ b “ is the base and “ h “ is the height of the triangular prism. Let us consider an example.

Suppose wish to find the surface area of a triangular prism having side 7 cm, base 5 cm and height 6 cm. The figure below illustrates the dimensions of such a prism –

The total surface area of this triangular prism will be given by – ab + 3bh

Here, a = 12 cm, b = 8 cm and h = 3 cm

Substituting these values in the given formula, we will have,

Total surface area of this triangular prism =12 x 8 + 3 x 8 x 3 = 96 + 72 = 168 sq. cm

**Using nets to find the surface area of a triangular prism**

Let us now consider an example, of how to find the surface area of a triangular prism using nets.

**Example**

Find the surface area of the following triangular prism using nets –

**Solution**

We can see that we have been given the length, bases and height of the triangular prism. Let us first find the total surface area of the triangular prism using the formula. Since we have been given all the three sides of the triangle, therefore, we will use the following formula to obtain the surface area –

Surface area of a triangular prism = ( S_{1} +S_{2} +S_{3} ) x l + b x h where

S_{1} , S_{2} and S_{3} are the three sides of the triangle, “ l “ is the length, “ b “ is the base and “ h “ is the height.

Surface area of a triangular prism = ( 8 + 10 + 10 ) x 17 + 8 x 6 = 476 + 48 = 524 sq. cm

Let us now find out the surface area of the triangular prism using nets.

The first step would be to unfold the given triangular prism in the form of a net. We would obtain the following on doing so –

It can be clearly seen that the net of a cube is made from 2 triangles and 3 rectangles. We will have,

Now,

Area of rectangle 1 = 17 x 10 = 170 sq. units

Area of rectangle 2 = 17 x 10 = 170 sq. units

Area of rectangle 3 = 17 x 8 = 136 sq. units

Area of triangle 1 4 = 12 x 8 x 6 = 24 sq. units

Area of triangle 2 = 12 x 8 x 6 = 24 sq. units

Hence, the total surface area of the given rectangular prism will be 170+ 170 + 136 +24 + 24 = 524 sq. units.

We can see that the result obtained by using the formula and the one obtained using the nets is the same.

In this way, we can use to find surface areas of different 3 – dimensional figures.

**Key Facts and Summary**

- 3 – dimensional shapes or 3D shapes are the shapes that have all the three dimensions, i.e. length, breadth and height.
- The surface area of any given object is the area or region occupied by the surface of the object.
- A cuboid whose length, breadth and height are equal is called a cube.
- A rectangular prism is a polyhedron with two congruent and parallel bases.
- A solid whose lateral faces are rectangular and the bases are congruent triangles is called a triangular prism.

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