Surface Area of a Rectangular Prism or Rectangular Solid
Where: H = height of rectangular prism
W = width of rectangular prism
L = length of rectangular prism
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A rectangular prism (also sometimes referred to as a "cuboid") is a three-dimensional figure consisting of six rectangular (and sometimes square)-shaped sides that are perpendicular to the other sides they touch.
Think of it this way, a rectangular prism is a hollow or solid three-dimensional figure with straight edges. It's a 3-D rectangle. Textbooks, rectangular table tops, most boxes, computer towers, and bricks are some common rectangular prisms. They are similar to cubes, but their faces are not all the same size as a cube's faces are.
To find the surface area of a rectangular prism, find the area of all six faces and add them together.
How the Equation Works
Each of the six faces of a rectangular prism is the same size as the face directly across from it.
Since most rectangular prisms have three different sizes of faces and there are two of each of these sizes, we must find these three sizes and multiply them each by two.
To find the value of each of the three sizes, we must find their areas. This is done by multiplying the perpendicular sides of each size together. So, since each of the three different sizes are made up of the shapes height and width, length and width, or length and height, each of these components must all be multiplied together.
With all of the areas of the three different sizes, we know to multiply them each by two (because there are two of each.) After that, all values for each of the six sides will be accounted for and they must be added together (because a shape's surface of area is the area of each of its sides added together.)
Uses
Knowing the surface area of a rectangular prism can be useful in a variety of situations. For example, if you're not sure if a rectangular prism-shaped object will fit in a certain box, you can calculate the surface area of the object and then the surface are of the box. If the surface area of the object is smaller than the surface area of the box, the object will fit inside of it.
Knowing whether or not an object will fit inside a box before you physically try to put the object in the box can save wasted time and energy, especially if the object is heavy.
Also, say you need to break down a box and use it to cover a portion of your floor (maybe you have a puppy that isn't potty-trained or likes to sleep on cardboard.) You know that you want to cover at least 7 square feet of floor. If you find the surface area of the box before you break it down, you'll know whether or not it's the size you're looking for.
Example
Charley needs a 175 in² portion of cardboard for a school project. He finds a box that's 5 inches high, 7 inches wide, and 3 inches long. When broken down, will this box be enough cardboard?
Charley needs to find the surface area of the box he found to see if it will supply enough cardboard. To do this he'll use our equation:
Surface area = 2HW + 2LW + 2LH
and then plug in the values for an answer.
H = 5 in W = 7 in L = 3 in
With those values, our equation becomes:
2(5)(7) + 2(3)(7) + 2(3)(5)
Now, just do the multiplication:
5 * 7 = 35
3 * 7 = 21
3 * 5 = 15
Which makes our formula:
2(35) + 2(21) + 2(15)
More multiplication:
2 * 35 = 70 in
2 * 21 = 42 in
2 * 15 = 30 in
Gives us:
(70) + (42) + (30)
Now we add these together and remember our units:
30 + 70 = 100
100 + 42 = 142 in²
This box will give him 142 square inches of cardboard. He needs 175. That means, with this box, he still needs (175 - 142 = 33) 33 square inches.
He'll have to find a bigger box or just find a smaller box to go with the one he already has.
A Little Bit About Square Units
Note that I kept track of the units and instead of inches, my answer was in square inches (square in, sq in, or in²). When calculating the surface area of a rectangular prism, the units of your answer will always be squared (square feet, square inches, etc.) This is because when assigning one number for the measurement of a rectangle, both the length and the width need to be accounted for.
By using square units you say how many single-unit squares fit into the larger shape being measured. So, for example, when I say the surface area of a rectangle prism is 150 sq in, it means that 150 small squares with the area of 1 sq in fit into the total area of the surface of our rectangular prism. This way, both the length and the width of the shape is included in your answer.
Drawing a Rectangular Prism
"Ok," you're asking, "I understand how to calculate its surface area, but is there some quick and simple way to draw it by hand?"
Of course there is. First, you start with two rectangles lined up like this:
Note: you can draw the bottom-right corner of the top rectangle to make your rectangular prism look like the one at the very top of this document or you can hide it to make it look slightly different.
Then, you connect the following corresponding corners with one straight line: top-right, top-left, bottom-left, and bottom-right (if the bottom rectangle isn't hiding the top rectangle's bottom-right corner.)
Related Websites:
http://www.math.com/tables/geometry/surfareas.htm
Works Cited
Rockswold, Gary, John Hornsby, and Margaret L. Lial. Precalculus Through Modeling
and Visualization. USA: Addison-Wesley Longman, Inc, May 2000.
"Cuboid." Wikipedia. 23 June 2006. Wikimedia Foundation, Inc. 30 June 2006.
<http://en.wikipedia.org/wiki/Rectangular_prism>.
"Surface Area Formulas." Math.com. 2005. Math.com. 29 June 2006.
<http://www.math.com/tables/geometry/surfareas.htm>.
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