A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which students have to choose the correct value, from a list of four, for the surface area of a rectangular prism whose length, width, and height are given.
There’s a diagram, and it looks something like this:
Unfortunately, the formula for surface area is not on the formula sheet provided by the state for this particular test. Therefore, we’re going to have to figure it out (if we don’t have it memorized).
To start with, a rectangular prism has six faces. The total surface area is just the area of each of those faces added up. And we know each face is a rectangle.
The formula for the area of a rectangle, of course, is length × width. Multiplying the two sides of each face by each other will give us the area for that face.
In this cube, the two bases (3 feet by 3 feet) have an area of 9 square feet. That makes 18 square feet for the two bases, which are actually squares in this case.
Next, each of the sides has a length of 15 feet and a width of 3 feet. Multiplying those gives us 45 square feet for each face on the side of the rectangular prism.
There are four of those faces, so 45 × 4 = 180.
Adding that to the 18 square feet we got for the combined areas of the two bases, we find that the total surface area of this rectangular prism is 198 square feet.
And finally, if you would like more practice on surface area problems, visit our online library at VoxLearn.org and search for “surface area.”
Try a few on your own
(1) Before they start their halftime show, members of the XXX High School marching band from XXX hide under a rectangular prism that measures one yard high, 20 yards on one side, and 20 yards on the other.
If each band member requires two cubic yards to hide under the prop, how many band members can fit under it at the same time?
(2) If the band wanted to cover the top of the prop, as well as the sides, with material, how many square feet of material would it take to completely cover the top and all four sides of the prop described above?
(3) If material at the fabric store costs $4.95 a yard, and it is two yards wide at the store, how much will it cost the band to buy material for the prop, as covered in Question (2) above?
The Answers
(1) First we have to determine the volume of the recanglar prism described in the question. To do this we know that the formula for the volume of a rectangular prism is given by the formula
where h is the height of the prism, l is the length, and w is the width. Thus we have
Then, we are told that each band member requires two cubic yards of space to fit under the prop. We have a total of 400 cubic yards of volume within the structure, divided by 2 cubic yards per person, gives us 200 people.
Our conclusion is that up to 200 band members would fit underneath the structure to make a spectacular entrance at the beginning of their halftime show.
(2) Hypothetically, if they wanted to cover the top with material, this is a question involving surface area. Except, we don’t need any material to cover the bottom, so it’s not really the whole surface area, is it?
Anyway, the area of the top is l × w, or 20 × 20, which gives us 400 square yards.
To that we have to add the area of each of the sides: 20 × 1 (length or width times the height) gives us 20 square yards for each of the sides. A rectangular prism has two bases and four other sides, so those four sides here would give us a total area of 80 (20 × 4) square yards.
The total surface area, then, of the one base (the top) plus the four sides, would be 400 + 80 = 480 square yards.
(3) In order to buy 480 square yards of material from the fabric store, we need to know how wide the fabric is. We are told the fabric comes with a width of two yards. How much of it do we need?
For this, we have to solve an equation. If the total area = length times width, our problem here is figuring out how much “length” will give us a certain total area if we know the “width”:
Plugging in the numbers, we have
We would need to buy 240 yards of the material, which costs $4.95 a yard at the store. 240 yards × $4.95/yard = $1,188.00. Schools don’t have to pay tax, so that would be the total cost of the material for the band.
Does it make sense to you that the top of the prop was left uncovered?
Illinois Alignment
Illinois Assessment Framework: 7.8.04 (8th grade) Solve problems involving the volume or surface area of a right rectangular prism, right circular cylinder, or composite shape using an appropriate formula or strategy.
Illinois Learning Standard: 7.C.3b (middle school) Use concrete and graphic models and appropriate formulas to find perimeters, areas, surface areas and volumes of two- and three-dimensional regions.