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7.02 Surface area of prisms

Lesson

Use of net to find the surface area

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

To find the surface area of a prism, we need to determine the kinds of areas we need to add together.

Consider this cube:

A cube with side length of 4. We can see the front, side and top faces.

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.

A net is has 6 identical squares: a row of 4 squares, 1 square at the bottom, and 1 square at the top of the second square.

By drawing the net of the cube we can see all the faces at once.

Now we know that the surface 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 net is useful for seeing exactly what areas need to be added together, but it isn't always this easy to find.

Another way to calculate the surface area of a prism is to calculate all the areas from the dimensions of the prism, without worrying about the exact area of each face.

Since prisms always have two identical base faces and the rest of the faces are rectangles connecting the two bases, we can accurately determine the dimensions of all the faces of a prism from just the dimensions of the base and the height of the prism.

In fact, we can think of all the rectangular faces joining the base faces as a single rectangle that wraps around the prism. One dimension of this rectangle must be the height of the prism. The other dimension of this rectangle will be the perimeter of the base.

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

This rectangular prism has dimensions of 8,\,7 and 5.

We choose the top and bottom faces to be the bases, and they each have areas of 8\times 7 =56.

To find the area of the rectangular faces joining the base faces, we multiply the height of the prism by the perimeter of one of the bases.

With two sides of length 8 and two sides of length 7, the base has a perimeter of 8+7+8+7=30, and multiplying by the height gives us the area 5\times 30 =150.

Adding this area to two copies of the base area tells us the total surface area for the prism: A=2 \times 56 + 150 = 262

We could instead find the area of each of the six rectangles and add them together, but using the perimeter can make some calculations faster.

Exploration

The following applet demonstrates how to find the surface area of various prisms using a formula.

Loading interactive...

As we move the sliders the only the dimensions are changing and not the number of faces of the prism.

Examples

Example 1

A rectangular prism with width of 5 metres, length of 7 metres, and height of 15 metres.

Consider the rectangular prism with a width, length and height of 5\text{ m},\,7\text{ m}, and 15\text{ m} respectively.

Find the surface area.

Worked Solution
Create a strategy

We need to add the areas of the 3 pairs of equal rectangular faces: Front and back, the left and right sides, and the top and bottom.

4 rectangular prisms showing the pairs of equal sides that make up the surface area. Ask your teacher for more information.
Apply the idea
\displaystyle \text{Front and back}\displaystyle =\displaystyle 2 \times 15 \times 5Multiply the area by 2
\displaystyle =\displaystyle 150\text{ m}^2Evaluate
\displaystyle \text{Sides}\displaystyle =\displaystyle 2 \times 15 \times 7Multiply the area by 2
\displaystyle =\displaystyle 210\text{ m}^2Evaluate
\displaystyle \text{Top and bottom}\displaystyle =\displaystyle 2 \times 7 \times 5Multiply the area by 2
\displaystyle =\displaystyle 70\text{ m}^2Evaluate
\displaystyle \text{Surface area}\displaystyle =\displaystyle 150+210+70Add the areas
\displaystyle =\displaystyle 430\text{ m}^{2}Evaluate

Example 2

Find the surface area of the figure shown. Give your answer to the nearest two decimal places.

A trapezoidal prism with dimensions of 10 cm, 5 cm, 13 cm, and 3 cm. Ask your teacher for more information.
Worked Solution
Create a strategy

Add the areas of the 6 faces together where the front and back face are identical trapeziums.

The image shows 4 trapezoidal prisms with different shaded faces. Ask your teacher for more information.
Apply the idea

We can find the area of each face using the following formulas:

\displaystyle \text{Front and back}\displaystyle =\displaystyle \dfrac{1}{2} (a+b)h \times 2Write the formula
\displaystyle =\displaystyle \dfrac{1}{2} \times (10 + 13) \times 5 \times 2 Substitute the dimensions
\displaystyle =\displaystyle 115\text{ cm}^{2}Evaluate
\displaystyle \text{Left rectangle}\displaystyle =\displaystyle 5 \times 3Multiply the length and width
\displaystyle =\displaystyle 15\text{ cm}^2Evaluate
\displaystyle \text{Top rectangle}\displaystyle =\displaystyle 10 \times 3Multiply the length and width
\displaystyle =\displaystyle 30\text{ cm}^2Evaluate
\displaystyle \text{Bottom rectangle}\displaystyle =\displaystyle 13 \times 3Multiply the length and width
\displaystyle =\displaystyle 39\text{ cm}^2Evaluate

We can find the missing dimension of the sloped side using Pythagoras' theorem:

\displaystyle c^2\displaystyle =\displaystyle a^{2}+b^{2}Write the formula
\displaystyle c^2\displaystyle =\displaystyle =\left(13-10\right)^2+5^2Substitute a and b
\displaystyle c^2\displaystyle =\displaystyle 34Evaluate
\displaystyle c\displaystyle =\displaystyle \sqrt{34}Square root both sides
\displaystyle =\displaystyle 5.830952Evaluate
\displaystyle \text{Right rectangle}\displaystyle =\displaystyle 5.830952 \times 3Multiply the length and width
\displaystyle =\displaystyle 17.49\text{ cm}^{2}Evaluate and round
\displaystyle \text{Surface area}\displaystyle =\displaystyle 115+15+30+39+17.49Add the areas
\displaystyle =\displaystyle 216.49\text{ cm}^{2}Evaluate
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.

Outcomes

MA5.1-8MG

calculates the areas of composite shapes, and the surface areas of rectangular and triangular prisms

MA5.2-11MG

calculates the surface areas of right prisms, cylinders and related composite solids

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