How to Easily Calculate the Surface Area of a Rectangular Prism

Rectangular prism (or cuboid) is the name for a six-sided, three-dimensional shape—also known as…a box! Picture a brick, a pair of game dice, or a shoebox, and you know exactly what a rectangular prism is. The surface area of a rectangular prism is the amount of space on the outside of the shape (or box), which can be calculated with the formula SA = 2 (wl + hl + hw). But we’ll break this formula down for you and take you step by step through the process of finding the surface area for any rectangular prism.

Surface Area for Rectangular Prism: Quick Steps

Multiply width by length, height by length, and height by width. Add all 3 numbers together, then multiply by 2 to get the surface area of the rectangular prism. The formula is SA = 2 (wl + hl + hw) where l = length, w = width, and h = height.

Section 1 of 4:

How to Calculate the Surface Area of a Rectangular Prism

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Draw the prism and label its length, width, and height, if necessary. If your prism isn’t already labeled, draw the 3D shape on a piece of paper. Write the symbols l, w, and h next to three different edges of the prism to represent length, width, and height, respectively.
- 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.
- 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.^{[1]
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Calculate the area of each face by multiplying its two edges together. A rectangular prism has six “faces:” a top and bottom face, a front and back face, and a left and right face. Each face of a rectangular prism is just a rectangle so, to find the area of each rectangle, “all you have to do is multiply length by width,” instructs expert academic tutor David Jia.
- In our example, l = 4, w = 3, and h = 5. These are the calculations you would do to find the correct area of each face:
  - Top face l x w = lw = 4 x 3 = 12
  - Bottom face l x w = lw = 4 x 3 = 12
  - Front face: w x h = wh = 3 x 5 = 15
  - Back face: w x h = wh = 3 x 5 = 15
  - Left face: l x h = lh = 4 x 5 = 20
  - Right face: l x h = lh = 4 x 5 = 20
- Alternatively, you can make a net of the rectangular prism to find its surface area. Making a net of a prism involves drawing out the shape as if it was two-dimensional—like the shape you would get if you broke down a box. Then, you can visually see each rectangle and calculate its area in the same way.
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Add the six areas together to get the value of the surface area. Add the area of each of the six faces together to get the surface area of the whole prism: lw + lw + wh + wh + lh + lh. You can use this formula for any rectangular prism to find the total surface area.^{[2]
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- To finish our example, just add up all of the areas you calculated in the previous step (aka blue numbers): 12 + 12 + 15 + 15 + 20 + 20 = 94.
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Add the correct unit measurements to the surface area to get your final answer. Surface area is measured in square units, e.g., mm², cm², in², ft², etc. Match your surface area measurements to the measurements used to describe the prism’s width, length, and height in the initial problem.^{[3]
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- For example, if you measured your rectangular prism’s length, width, and height in inches (i.e., l = 4 in, w = 3 in, and h = 5 in), then your final answer must also use inches (i.e., 94 square inches (in²)).
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To calculate SA more quickly, use the formula SA = 2 (wl + hl + hw). If you don’t have to show your work solving for surface area, you can skip most of the above steps and simply plug the width, length, and height of the rectangular prism into this formula.^{[4]
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- For example, a prism with l = 4, w = 3, and h = 5 would generate this formula: SA = 2 ((3x4) + (5x4) + (5x3)) = 2 (12 + 20 + 15) = 2 (47) = 94.
- If you want to test out how this formula works, start with the formula we used before to calculate the surface area with the areas of each face: lw + lw + wh + wh + lh + lh.
- If you combine all of the like terms, you get 2lw + 2wh + 2lh. Then you can factor out the two to make it even shorter, ending up with 2(lw + wh + lh).

Section 2 of 4:

Surface Area of a Rectangular Prism: Example Problems

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Problem 1: SA of a Rectangular Prism Calculate the surface area of a rectangular prism with the measurements l = 9cm, w = 3cm, and h = 4cm. Solution:
- Calculate the area of each face:
  - Top & Bottom: 9 x 3 = 27 cm²
  - Front & Back: 9 x 4 = 36 cm²
  - Left & Right: 3 x 4 = 12 cm²
- Add the six areas together: 27 + 27 + 36 + 36 + 12 + 12 = 150.
- Include the units: The initial measurements on the rectangular prism are in cm, meaning the final surface area of the rectangular prism is 150 cm².
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Problem 2: SA of a Rectangular Prism Calculate the surface area of a rectangular prism with the measurements l = 2mm, w = 10mm, and h = 5mm. Solution:
- Calculate the area of each face:
  - Top & Bottom: 2 x 10 = 20 mm²
  - Front & Back: 2 x 5 = 10 mm²
  - Left & Right: 5 x 10 = 50 mm²
- Add the six areas together: 20 + 20 + 10 + 10 + 50 + 50 = 160.
- Include the units: The initial measurements on the rectangular prism are in mm, meaning the final surface area of the rectangular prism is 160 mm².
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Problem 3: SA of a Cube Calculate the surface area of a cubic rectangular prism with the measurements l = 8in, w = 8in, and h = 8in. Solution:
- Calculate the area of each face: Each face of a cube is the same, so the area of each face is 8 x 8 = 64 in².
- Add the six areas together: 64 + 64 + 64 + 64 + 64 + 64 = 384.
- Include the units: The initial measurements on the rectangular prism are in inches, meaning the final surface area of the rectangular prism is 384 in².

Section 3 of 4:

Common Mistakes When Calculating Surface Area

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It can be easy to confuse volume and surface area when you’re first learning geometry, but these are two different concepts. The volume of a prism is the space inside the shape and is measured in cubic units (e.g., cm³). The surface area of a prism is the total area of the faces on the outside of the shape and is measured in square units (e.g., cm²).^{[5]
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Assuming all lateral faces are equal Unless your rectangular prism is a cube, only the opposite sides of a shape will be equal. Essentially, the top and bottom of the prism may be equal to each other, but the other sides will have a different measurement.^{[6]
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- It’s possible for all lateral faces to be equal, but only if the base is a square.

Section 4 of 4:

Calculating the Volume & Space Diagonal of a Rectangular Prism

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Calculating the volume of a rectangular prism To calculate the volume of a rectangular prism, plug the shape’s width (w), length (l), and height (h) into the formula $V=whl$ $V=whl$ .^{[7]
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- For example, a rectangular prism with the measurements l = 2cm, w = 3cm, and h = 4cm will have Volume = 3 x 4 x 2 = 24 cm³.
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Calculating the space diagonal of a rectangular prism To calculate the space diagonal of a rectangular prism, plug the shape’s width (w), length (l), and height (h) into the formula $d=sqrt(l^{2}+w^{2}+h^{2})$ $d=sqrt(l^{2}+w^{2}+h^{2})$ .^{[8]
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- For example, a rectangular prism with the measurements l = 2cm, w = 3cm, and h = 4cm will have space diagonal = sqrt(2²+3²+4²) = 5.38516.

Surface Area Calculator & Cheat Sheets

Surface Area of a Rectangular Prism Using Dimensions Cheat Sheet

Surface Area of a Rectangular Prism Using Area Cheat Sheet

Surface Area of a Rectangular Prism Calculator

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

Donagan

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Multiply x by the width and then by the height.

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

Donagan

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You can't find them without having additional information.

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

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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.

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In U.S. states that follow the Common Core curriculum, finding the surface area of three-dimensional figures (like rectangular prisms) is taught in 6th grade math as part of their geometry studies.^{[9]
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If you’re teaching a student how to find the surface area of a rectangular prism, try to connect the problems to real-life applications, like wrapping gifts or painting the sides of a prism.

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