From this angle we can see three square faces with side length 4, and the area of these faces will contribute to the surface area. But we also need to consider the faces we can't see from this view.

Exploration

Let's explore the following applet to find the surface area of a rectangular prism with its net.

Let's start with the cube above with side length 4\operatorname{cm}.

Move the points to adjust the length, the width and height of the prism to be equal to 4\operatorname{cm}.
Tick "Show net".
Tick "Show surface area"
.

Loading interactive...

How many faces does the cube have?

What shape/shapes composed this cube?

How do you get the surface area of this cube?

Now we know that the surface of a cube is made up of six identical square faces, and finding the surface area of the cube is the same as finding the area of a square face and multiplying that by 6: A=6 \times 4^{2}=96

Using the applet above, adjust the length, width and height of the prism.

What is the effect of increasing the length, width or height on the surface area of the prism?

Are there pairs of congruent faces?

How do you get the surface area of this prism?

Examples

Example 1

Consider the following rectangular prism:

A rectangular prism with height of 5 units,length of 7 units and width of 8 units.

Which of the following nets match the given rectangular prism?

A net with given dimension.Speak to your teacher for more information.

Worked Solution

Create a strategy

How many faces does a closed rectangular prism have?

Consider what the dimensions of those faces should be.

Apply the idea

The closed rectangular prism has 6 faces. Considering the dimensions, the nets should be option C.

Find the surface area of the rectangular prism.

Worked Solution

Create a strategy

The surface area is the sum of all the faces of the prism. The surface area of a solid is equal to the area of the solid's net.

Apply the idea

The net of a rectangular prism is made up of three pairs of equal rectangles, with each pair representing one possible pair of dimensions of the prism.

Since this rectangular prism has dimensions of 8cm , 7cm and 5cm, the pairs of dimensions for the faces of the net will be 8\times 7, 8\times 5 and 7\times 5.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2\times 8\times 7 + 2\times 8\times 5 + 2\times 7 \times 5 \text{ cm}^2
	\displaystyle =	\displaystyle 112+80+70	Evaluate the products.
	\displaystyle =	\displaystyle 262\text{ cm}^2	Evaluate the sum.

The surface area of the rectangular prism is 262\text{ cm}^2.

Idea summary

The surface area of a prism is the sum of the areas of all the faces.

Drawing the net of a prism is useful for seeing exactly what areas need to be added together.

Surface area with nets of pyramids

For pyramids, we should remember that they have triangular sides, and the shape of the base gives the prism its name.

The surface area of a right pyramid is the sum of the area of the base and the area of the triangles.

This image shows a net of a solid. Ask your teacher for more information.

How many faces does this pyramid have?

What are the shapes of its faces?

How do we get the surface area of this pyramid?

The net shows the 5 faces that make up this pyramid.

There are 1 square base and 4 triangular faces.

To find the surface area of the pyramid, we'll add the areas of the base and its faces.

Examples

Example 2

Find the surface area of the following pyramid.

A rectangular pyramid with given dimension. Ask your teacher for more information.

Worked Solution

Create a strategy

Remember the surface area of a rectangular pyramid consists of four triangles and a rectangle.

Each slant height of the rectangular pyramid will correspond to the height of one of the pairs of identical triangle faces.

Apply the idea

We can calculate the area of each face of the pyramid using the equations:

\displaystyle \text{Area of rectangle}

\displaystyle =

\displaystyle \text{Length}\times \text{Width}

\displaystyle \text{Area of triangle}

\displaystyle =

\displaystyle \dfrac{1}{2}\times \text{Base} \times \text{Height}

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle \text{Area of the rectangle base} + \text{Area of the triangles}
	\displaystyle =	\displaystyle 8 \times 7 + 2 \times \dfrac{1}{2} \times 7 \times 8 + 2\times \dfrac{1}{2} \times 8 \times 10 \text{ cm}^2
	\displaystyle =	\displaystyle 56+56+80 \text{ cm}^2	Evaluate the products.
	\displaystyle =	\displaystyle 192\text{ cm}^2	Evaluate the sum.

Idea summary

The surface area of a pyramid is the sum of the areas of the base and the triangular faces.

The slant height of the triangle is the height of the triangle.

Outcomes

6.G.A.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

7.06 Surface area with nets

Ideas

Surface area with nets of prisms

Exploration

Examples

Example 1

Surface area with nets of pyramids

Examples

Example 2

Outcomes

6.G.A.4

What is Mathspace

About Mathspace