how to find surface area of rectangular prism

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A rectangular prism is a name for a 6-sided 3-dimensional figure that is very familiar to everybody—a box.^[1] Think of a brick, or a shoebox, and you know exactly what a rectangular prism is. The surface area is the amount of space on the outside of the object. "How much paper do I need to wrap this shoebox" sounds a lot less complicated, but it's exactly the same math problem.

Surface Area Help

1
Label the length, width, and height of your rectangular prism. Each rectangular prism has a length, a width, and a height. Draw a picture of the prism, and write the symbols l, w, and h next to three different edges of the shape.
- If you're not sure which sides to label, pick any corner. Label the three lines that meet at that corner.
- For example: A box has a base that measures 3 inches by 4 inches, and it stands 5 inches tall. The long side of the base is 4 inches, so l = 4, w = 3, and h = 5.
2
Look at the six faces of the prism. To cover the whole surface area, you'd need to paint six different "faces." Think about each one — or find a box of cereal and look at them directly:
- There are a top and bottom face. Both are the same size.^[2]
- There are a front and a back face. Both are the same size.
- There are a left and right face. Both are the same size.
- If you have trouble picturing this, cut a box apart along the edges and lay it out.^[3]
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3
Find the area of the bottom face. To start out, let's find the surface area of just one face: the bottom. This is a rectangle, just like every face. One edge of the rectangle is labeled length and the other is labeled width. To find the area of the rectangle, just multiply the two edges together.^[4] Area (bottom edge) = length times width = lw.
- Going back to our example, the area of the bottom face is 4 inches x 3 inches = 12 square inches.
4
Find the area of the top face. Wait a second — we already noticed that the top and bottom faces are the same size. This must also have an area of lw.
- In our example, the top area is also 12 square inches.
5
Find the area of the front and back faces. Go back to your diagram and look at the front face: the one with one edge labeled width and one labeled height. The area of the front face = width times height = wh. The area of the back is also wh.
- In our example, w = 3 inches and h = 5 inches, so the area of the front is 3 inches x 5 inches =15 square inches. The area of the back face is also 15 square inches.
6
Find the area of the left and right faces. We've just got two faces left, each the same size. One edge is the length of the prism, and one edge is the height of the prism. The area of the left face is lh and the area of the right face is also lh.
- In our example, l = 4 inches and h = 5 inches, so the area of the left face = 4 inches x 5 inches = 20 square inches. The area of the right face is also 20 square inches.
7
Add the six areas together. Now you've found the area of each of the six faces. Add them all together to get the area of the whole shape: lw + lw + wh + wh + lh + lh. You can use this formula for any rectangular prism, and you will always get the surface area.
- To finish our example, just add up all the blue numbers above: 12 + 12 + 15 + 15 + 20 + 20 = 94 square inches.
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1
Simplify the formula. You now know enough to find the surface area of any rectangular prism. You can do it faster if you've learned some basic algebra. Start with our equation above: Area of a rectangular prism = lw + lw + wh + wh + lh + lh. If we combine all the terms that are the same, we get:
- Area of a rectangular prism = 2lw + 2wh + 2lh
2
Factor out the two. If you know how to factor in algebra, you can make it even shorter:
- Area of a Rectangular Prism = 2lw + 2wh + 2lh = 2(lw + wh + lh).
3
Test it on an example. Let's go back to our example box, with length 4, width 3, and height 5. Plug these numbers into the formula:
- Area = 2(lw + wh + lh) = 2 x (lw + wh + lh) = 2 x (4x3 + 3x5 + 4x5) = 2 x (12 + 15 + 20) = 2 x (47) = 94 square inches. That's the same answer we got before. Once you've practiced doing these equations, this is a much faster way to find the surface area.
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Question

How do I find the surface area of one with no length or is represented by x?

Multiply x by the width and then by the height.
Question

How do I find the edge lengths for a rectangular prism with a surface area of 92 m?

You can't find them without having additional information.
Question

How do I find the total surface area of a triangular prism?

Start off with the formula for the area of a triangle: 1/2bh = a (One half of base times height equals area.) Also, you'll need to know how to find the area of a rectangle, lw = a (length times width equals area.) Make a net of the prism. If the length and width of the prism are say, l = 4 and w = 6, the bottom rectangle in the center should be 4 x 6 (area = 24 sq. units.). Next, do the other two rectangles (Cheat: They're always the same area as the base!) Now, find the area of the triangle. Say the height = 4. We know w = 6, so we multiply 4 x 6. Now we multiply that by 1/2 (divide by 2). Do the same for the other one, then add them up.
Question

How do I find the surface area of a triangular pyramid?

Find the surface area of each triangle (base x height), then add them together. If all the triangles are congruent, you can just find the surface area of one and multiply it by 4.
Question

What is the measure of each edge of a rectangle if the surface area is 600?

Without knowing more about the rectangular prism and the ratio of the side measurements, it would be difficult to calculate. However, if you are talking about a cube in particular, every side would measure 10 units, if the surface area is 600 units squared.
Question

If the surface area, length, and width are given, how do I determine the height of a rectangular prism?

Multiply length by width, and double it. That gives you the total area of the top and bottom surfaces of the prism. Subtract that from the total surface area you were given. That leaves you with the total area of the four sides of the prism. Set that number aside for a moment. Add the length and width together and double it. That gives you the perimeter around either the top or bottom surface. Divide that perimeter into the total area of the four sides you figured out a moment ago. That gives you the height.
Question

How do I find the length (base) of a rectangular prism if I am given the height, width, and total surface area?

Multiply the width by the height. Double that number. That gives the total surface area of the two smaller ends of the prism. Subtract that figure from the total surface area of the prism. That leaves you with the total area of the other four surfaces. Let's call them the four main surfaces. Add the height to the width and double that figure. That gives you the perimeter around the four main surfaces. Divide that perimeter into the total area of the four main surfaces . That gives you the length of the prism.
Question

How do I write an equation representing the area of a rectangle?

Area equals length multiplied by width.
Question

How do I determine the area of the left side of a triangular prism when only given the area of the rectangle in front?

You can't without more information.
Question

What is the surface area of a right rectangular prism with dimensions of 4 cm by 4 cm by 6 cm?

top and bottom: 16 cm^2 ea. side front and back: 16cm^2 ea. side side and side: 24cm^2 ea. side Total surface area = 112 cm^2

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Areas always use "square units," like square inches or square centimeters.^[5] A square inch is just what it sounds like: a square that's one inch wide and one inch long. If a prism has a surface area of 50 square inches, that means it takes 50 of those squares to cover every surface on the prism.
Some teachers use "breadth" or "depth" instead of one of these names. That's fine, as long as you label each side clearly.
If you don't know which way up the prism is, you can call any side the height. The length is usually the longest side, but even that's not really important. As long as you stick with the same names for the whole problem, you'll be fine.^[6]

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Article SummaryX

To find the surface area of a rectangular prism, measure the length, width, and height of the prism. Find the area of the top and bottom faces by multiplying the length and width of the prism. Then, calculate the area of the left and right faces by multiplying the width and height. Finally, find the area of the front and back faces by multiplying the length and height of the prism. To find the surface area, simply add all 6 of these areas together and write your result in square units. If you want to learn how to simplify your formulas to make them easier to remember, keep reading the article!

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