9.02 Surface area of prisms

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:

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.

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.

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.

Examples

Example 1

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.

Apply the idea

\displaystyle \text{Front and back}	\displaystyle =	\displaystyle 2 \times 15 \times 5	Multiply the area by 2
	\displaystyle =	\displaystyle 150\text{ m}^2	Evaluate
\displaystyle \text{Sides}	\displaystyle =	\displaystyle 2 \times 15 \times 7	Multiply the area by 2
	\displaystyle =	\displaystyle 210\text{ m}^2	Evaluate
\displaystyle \text{Top and bottom}	\displaystyle =	\displaystyle 2 \times 7 \times 5	Multiply the area by 2
	\displaystyle =	\displaystyle 70\text{ m}^2	Evaluate
\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 150+210+70	Add the areas
	\displaystyle =	\displaystyle 430\text{ m}^{2}	Evaluate

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Example 2

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

Worked Solution

Create a strategy

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

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 2	Write 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 3	Multiply the length and width
	\displaystyle =	\displaystyle 15\text{ cm}^2	Evaluate
\displaystyle \text{Top rectangle}	\displaystyle =	\displaystyle 10 \times 3	Multiply the length and width
	\displaystyle =	\displaystyle 30\text{ cm}^2	Evaluate
\displaystyle \text{Bottom rectangle}	\displaystyle =	\displaystyle 13 \times 3	Multiply the length and width
	\displaystyle =	\displaystyle 39\text{ cm}^2	Evaluate

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+5^2	Substitute a and b
\displaystyle c^2	\displaystyle =	\displaystyle 34	Evaluate
\displaystyle c	\displaystyle =	\displaystyle \sqrt{34}	Square root both sides
	\displaystyle =	\displaystyle 5.830952	Evaluate
\displaystyle \text{Right rectangle}	\displaystyle =	\displaystyle 5.830952 \times 3	Multiply the length and width
	\displaystyle =	\displaystyle 17.49\text{ cm}^{2}	Evaluate and round

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 115+15+30+39+17.49	Add the areas
	\displaystyle =	\displaystyle 216.49\text{ cm}^{2}	Evaluate

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