7.01 Surface area of rectangular prisms

A rectangular prism is a polyhedron in which all six faces are rectangles. A rectangular prism has 8vertices and 12 edges.

The surface area of a rectangular prism is the sum of the area of all of its faces. We can use a net to visualize the prism as a flat, 2D figure and easily calculate its surface area.

Net

A two-dimensional representation of a three-dimensional figure that can be folded into a model of the three-dimesional figure.

Rectangular prisms have three pairs of congruent faces. We can see below how we could break the rectangular prism above into three pairs of congruent rectangles. To find the total surface area, we must add up the area of all of the faces.

\text{Surface area of a prism} = \text{Sum of areas of faces}

We can also use a formula instead of adding up all 6 faces separately.

A rectangular prism with length l, width w, and height h

The 2D net of the same prism

We can see the rectangular prism has three pairs of congruent rectangles.

• The top and bottom which are both l \cdot w

• The left and right which are l \cdot h

• The front and back which are w \cdot h

Since there are two of each of these rectangles we get the formula below.

SA=2lw+2lh+2wh

Examples

Example 1

Consider the following cube with a side length equal to 6 \text{ cm}.

Find the total surface area.

Worked Solution

Create a strategy

We can use the surface area of rectangular prism formula: SA=2lw + 2lh + 2wh, where \\l=\text{length},\, w=\text{width}, and h=\text{height}.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2(6\cdot 6) + 2(6 \cdot 6) + 2(6 \cdot 6)	Substitute the values of l, w, \text{and} \,h
	\displaystyle =	\displaystyle 2(36) + 2(36) +2(36)	Evaluate the multiplication inside the brackets
	\displaystyle =	\displaystyle 72 + 72 + 72	Evaluate the multiplication
	\displaystyle =	\displaystyle 216\text{ cm}^{2}	Evaluate the addition

Reflect and check

We can also use the surface area of cube formula: \text{Surface area}=6\cdot \text{side}^{2}, where 6=\text{no. of faces.}

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 6\cdot (6)^{2}	Substitute the values
	\displaystyle =	\displaystyle 6 \cdot 36	Evaluate the square
	\displaystyle =	\displaystyle 216\text{ cm}^{2}	Evaluate the multiplication

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Example 2

Consider the following rectangular prism with length, width and height equal to 12 \text{ m},\,6 \text{ m} and 4 \text{ m} respectively.

Find the surface area of the prism.

Worked Solution

Create a strategy

We can use the surface area of rectangular prism formula: SA=2lw + 2lh + 2wh, where \\l=\text{length},\, w=\text{width}, and h=\text{height}.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2(12\cdot 6) + 2(12 \cdot 4) + 2(6 \cdot 4)	Substitute the values of l, w, \text{and} \,h
	\displaystyle =	\displaystyle 2(72) + 2(48) +2(24)	Evaluate the multiplication inside the brackets
	\displaystyle =	\displaystyle 144 + 96 + 48	Evaluate the multiplication
	\displaystyle =	\displaystyle 288\text{ m}^{2}	Evaluate the addition

Reflect and check

This calculation accounts for all faces of the prism by adding the area of the front and back, the left and right, and the top and bottom faces, then doubling the result. You could also calculate each face separately and then add them together to verify that you get the same total surface area.

Rectangle 1:

\begin{aligned}\text{Area 1}=&l \cdot w\\=& 12 \cdot 6\\=& 72\text{ m}^2\end{aligned}

Rectangle 2:

\begin{aligned}\text{Area 2}=&l \cdot w\\=& 12 \cdot 6\\=& 72\text{ m}^2\end{aligned}

Rectangle 3:

\begin{aligned}\text{Area 3}=&l \cdot w\\=& 12 \cdot 6\\=& 72\text{ m}^2\end{aligned}

Rectangle 4:

\begin{aligned}\text{Area 4}=&l \cdot w\\=& 12 \cdot 6\\=& 72\text{ m}^2\end{aligned}

After we calculate the area of each of the rectangles, we can find the total surface area of the rectangular prism by adding them together.

\displaystyle \text{Total surface area}	\displaystyle =	\displaystyle \text{Area 1} + \text{Area 2} + \text{Area 3} + \text{Area 4}
	\displaystyle =	\displaystyle 72 + 72 + 72 + 72
	\displaystyle =	\displaystyle 288\text{ m}^2

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