In manufacturing we may need to calculate the cost of making boxes or sheet metal parts.
In construction, surface area affects planning (how much to buy) and costs (how much to charge) in connection with such things as wallboard, shingles, and paint.

Many objects have complex shapes to increase their surface area: the inside of your lungs, intestines, and brain; air purifiers, or radiators.

We will start by looking at how to find the surface area of a rectangular prism.

Visualizing surface area of rectangular prisms

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.

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Exploration

Have a look at this interactive to see how to unfold rectangular prisms.

Slide the "Open/Close" slider to see the prism fold and unfold
Slide one of the "Dimensions" sliders and then Open/Close again

Consider the questions below.

What type of shape is formed if all of the Dimension sliders are set to the same level?
How many faces does every rectangular prism have?
If you change only one of the Dimension sliders how many congruent faces are there?
What is the benefit to unfolding and opening the prism when calculating the surface area?
Instead of finding the area of each of the $6$6 faces, how could you find the surface area more quickly?

Credit: Tom Healy

Calculating surface area of rectangular prisms

When needing to calculate the surface area (SA) of a prism we need to add up the areas of individual faces. Make sure not to miss any faces but also try to look for clever methods, like using the fact that $2$2 faces might have the same area.

While we are just looking at rectangular prisms for now, the concept below will help us in future lessons too.

General: Surface area of a prism

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

If we are just looking at a rectangular prism, we can use a formula instead of adding up all $6$6 faces separately.

Rectangular prism with length $l$l, width $w$w and height $h$h

As we saw with the applet above, there are three pairs of congruent rectangles.

The top and bottom which are both $l\times w$l×w
The left and right which are $l\times h$l×h
The front and back which are $w\times h$w×h

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

Rectangular prism surface area formula

$\text{S.A. }=2lw+2lh+2wh$S.A. =2lw+2lh+2wh

Practice questions

QUESTION 1

Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.

A three-dimensional cube with single hash mark on each side of the cube. One of the side measures $6$ cm.

QUESTION 2

Consider the following rectangular prism with length, width, and height, equal to $12$12 m, $6$6 m, and $4$4 m, respectively.

Find the surface area of the prism.

A rectangular prism with length, width, and height equal to $12$ m, $6$ m, and $4$ m, respectively. It is oriented so that the length extends horizontally into the plane of the page, the width is horizontal and extends towards the right, and the height is vertical.

QUESTION 3

What is the surface area of a cube with side length $4$4 cm?

Outcomes

7.G.B.5

Solve real-world and mathematical problems involving area of two dimensional figures composed of triangles, quadrilaterals, and polygons, and volume and surface area of three-dimensional objects composed of cubes and right prisms.

7.07 Surface area of rectangular prisms