How to calculate the surface area of a prism

Square Pyramid Shape

h = height
s = slant height
a = side length
e = lateral edge length
r = a/2
V = volume
Stot = total surface area
Slat = lateral surface area
Sbot = bottom surface area

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Online calculator to calculate the surface area of geometric solids including a capsule, cone, frustum, cube, cylinder, hemisphere, pyramid, rectangular prism, sphere, spherical cap, and triangular prism

Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft2 or ft3. For example, if you are starting with mm and you know r and h in mm, your calculations will result with V in mm3 and S in mm2.

Below are the standard formulas for surface area.

Surface Area Formulas:

Capsule Surface Area

• Volume = πr2((4/3)r + a)
• Surface Area = 2πr(2r + a)

Circular Cone Surface Area

• Volume = (1/3)πr2h
• Lateral Surface Area = πrs = πr√(r2 + h2)
• Base Surface Area = πr2
• Total Surface Area
  = L + B = πrs + πr2 = πr(s + r) = πr(r + √(r2 + h2))

Circular Cylinder Surface Area

• Volume = πr2h
• Top Surface Area = πr2
• Bottom Surface Area = πr2
• Total Surface Area
  = L + T + B = 2πrh + 2(πr2) = 2πr(h+r)

Conical Frustum Surface Area

• Volume = (1/3)πh (r12 + r22 + (r1 * r2))
• Lateral Surface Area
  = π(r1 + r2)s = π(r1 + r2)√((r1 - r2)2 + h2)
• Top Surface Area = πr12
• Base Surface Area = πr22
• Total Surface Area
  = π(r12 + r22 + (r1 * r2) * s)
  = π[ r12 + r22 + (r1 * r2) * √((r1 - r2)2 + h2) ]

Cube Surface Area

• Volume = a3
• Surface Area = 6a2

Hemisphere Surface Area

• Volume = (2/3)πr3
• Curved Surface Area = 2πr2
• Base Surface Area = πr2
• Total Surface Area= (2πr2) + (πr2) = 3πr2

Pyramid Surface Area

• Volume = (1/3)a2h
• Lateral Surface Area = a√(a2 + 4h2)
• Base Surface Area = a2
• Total Surface Area
  = L + B = a2 + a√(a2 + 4h2))
  = a(a + √(a2 + 4h2))

Rectangular Prism Surface Area

• Volume = lwh
• Surface Area = 2(lw + lh + wh)

Sphere Surface Area

• Volume = (4/3)πr3
• Surface Area = 4πr2

Spherical Cap Surface Area

• Volume = (1/3)πh2(3R - h)
• Surface Area = 2πRh

Triangular Prism Surface Area

Top Surface Area of a Triangular Prism Formula

\[ A_{top} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]

\[ A_{top} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]

Bottom Surface Area of a Triangular Prism Formula

\[ A_{bot} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]

\[ A_{bot} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]

Lateral Surface Area of a Triangular Prism Formula

\[ A_{lat} = h (a+b+c) \]

Total Surface Area of a Triangular Prism Formula

\[ A_{tot} = A_{top} + A_{bot} + A_{lat} \]

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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. 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.[2]

   • 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. 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:[3]

   • There are a top and bottom face. Both are the same size.[4]
   • 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.[5]

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3. 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.[6] 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. 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[7].

   • In our example, the top area is also 12 square inches.

5. 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. 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. 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.[8]

   • 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. 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:[9]

   • Area of a rectangular prism = 2lw + 2wh + 2lh

2. 2

   Factor out the two. If you know how to factor in algebra, you can make it even shorter:[10]

   • Area of a Rectangular Prism = 2lw + 2wh + 2lh = 2(lw + wh + lh).

3. 3

   Test it on an example.[11] 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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Add New Question

• 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.

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• Areas always use "square units," like square inches or square centimeters.[12] 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.

  <b>Thanks! We’re glad this was helpful.</b><br />Looking for more fun ways to learn on wikiHow?<br />Learn about yourself with <a href='/Quizzes'><b>Quizzes</b></a> or try our brand new <a href='/Games/Train-Your-Brain'><b>Train Your Brain</b></a> word game.

• 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.[13]

  <b>Thanks! We’re glad this was helpful.</b><br />Looking for more fun ways to learn on wikiHow?<br />Learn about yourself with <a href='/Quizzes'><b>Quizzes</b></a> or try our brand new <a href='/Games/Train-Your-Brain'><b>Train Your Brain</b></a> word game.

• Some teachers use "breadth" or "depth" instead of one of these names. That's fine, as long as you label each side clearly.

  <b>Thanks! We’re glad this was helpful.</b><br />Looking for more fun ways to learn on wikiHow?<br />Learn about yourself with <a href='/Quizzes'><b>Quizzes</b></a> or try our brand new <a href='/Games/Train-Your-Brain'><b>Train Your Brain</b></a> word game.

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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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