How to Find the Surface Area of a Rectangular Prism

Surface area is how much material it takes to wrap something. Volume is how much fits inside. They are different questions with different units, and a box that holds a lot is not necessarily a box that costs a lot to make. Get into the habit of asking which one the problem actually wants, because reaching for the wrong one is the most common way this goes wrong before any arithmetic happens.

Six faces, three pairs

A rectangular prism is a box, and a box has six flat faces. They are not six different faces though. The front matches the back, the left matches the right, and the top matches the bottom, so there are only three distinct areas to work out. That is the entire reason the formula has a 2 outside the bracket instead of six separate terms.

Each of the three uses two of the three dimensions. Length times width gives the top, length times height gives the front, width times height gives the side. Every pairing of two dimensions appears exactly once, which is a useful check: if your three face areas do not use each dimension twice between them, you have measured something wrong.

The net is the whole explanation

Cut a cardboard box along its edges and flatten it. What you get is a net: six rectangles joined in a cross. The surface area of the box is just the area of that flat shape, which is why nothing more complicated than length times width is ever needed. Anyone who has taken apart a delivery box already knows this formula, they have just not written it down.

A worked check you can repeat on paper

Route three is worth keeping, because it is the one that generalises. Wrap a strip around the four sides and its area is the perimeter of the base times the height, then cap the ends with two copies of the base. That works for any prism at all, whatever shape the ends are, which is how you handle a triangular or hexagonal prism without learning a new formula for each.

Surface area and volume do not grow together

Double every dimension of the worked box, from 8 by 5 by 3 to 16 by 10 by 6. The surface area goes from 158 to 632, which is four times. The volume goes from 120 to 960, which is eight times. Area follows the square of the scale, volume follows the cube, and they part company immediately.

That gap has consequences well outside a maths exercise. It is why a large box costs less cardboard per litre of contents than a small one, why a big block of ice takes so much longer to melt than the same mass in cubes, and why large animals struggle to shed heat while small ones struggle to keep it. Make something bigger and its skin can never keep up with its bulk.

Shape matters as much as size. These three boxes all hold exactly 288 cubic units, and the material each one needs is not close:

Box	Volume	Surface area
12 by 12 by 2, a flat tray	288	384.0
8 by 6 by 6, a chunky box	288	264.0
6.60 cubed, as close to a cube as it gets	288	261.7

The flat tray needs 47 percent more material than the cube to hold the same contents. For a given volume the cube is always the cheapest box to make, and a sphere beats even that, which is why bubbles are round and why packaging designers fight to make boxes squarer. The worked example is not far off: a cube holding 120 would have sides of 4.93 and a surface area of 146, so the 8 by 5 by 3 shape wastes about 8 percent.

Mistakes that catch people out

• Calculating volume by accident. Multiplying all three dimensions at once gives 120, which is the volume. Surface area never multiplies three lengths together, only two at a time. If your answer came from one multiplication, it is the wrong quantity.
• Forgetting to double. Stopping at 79 is the single most common slip, and it is exactly half the right answer. The bracket holds one of each pair, not all six faces.
• Doubling the wrong thing. 2lw + lh + wh is not the same as 2(lw + lh + wh). The 2 applies to the whole bracket, so on paper it is safer to add first and double last.
• Giving the answer in cubic units. Surface area is measured in square units, because every term is one length times another. Only volume earns the cube.
• Counting faces that are not there. An open topped box has five faces, not six, and a length of guttering has four. Read the question before applying the formula, and subtract the missing face rather than starting over.

Try these

Common questions

What is the formula for the surface area of a rectangular prism?

SA = 2(lw + lh + wh). Work out the three different face areas, add them, then double, because each face has a matching one opposite. A box 8 by 5 by 3 gives 2(40 + 24 + 15), which is 158 square units.

What is the difference between surface area and volume?

Surface area is the material needed to cover the outside, measured in square units. Volume is the space inside, measured in cubic units. For the box above they are 158 and 120, two different numbers answering two different questions.

How do you find a missing dimension from the surface area?

Subtract the faces you can work out, then divide. With a surface area of 158, a length of 8 and a width of 5, the top and bottom account for 80, leaving 78 for the side strip. That strip is 26 wide, so the height is 78 over 26, which is 3.

What is a net?

The flat shape you get by cutting a solid along its edges and unfolding it. For a rectangular prism the net is six rectangles in a cross, and its area is the surface area of the box.

Why does doubling the size multiply the surface area by four?

Because every face is one length multiplied by another. Double both and that face becomes four times as big, so the whole surface does too. Volume multiplies three lengths, so it becomes eight times as big instead.

Which box shape uses the least material?

The cube, for any given volume. A flat 12 by 12 by 2 tray and a 6.6 sided cube both hold 288 cubic units, but the tray needs 384 square units of material against the cube’s 262, which is 47 percent more.

Related lessons and calculators

• Area formulas for every common shape, since every face of a solid is a flat shape that has to be measured first.
• Area of a parallelogram, where the base times height rule that every rectangular face relies on comes from.
• Volume of 3D shapes calculator, for the other question people usually need answered about the same solid.
• Area of 2D shapes calculator, useful when the ends of a prism are not rectangles.
• Paint coverage calculator, which is surface area put to work, since paint is sold against area and never against volume.
• NIST on SI units, the reference for why area is squared and volume is cubed.
• Surface area of a triangular prism, the same two ends and a wrap method when the end is not a rectangle.
• Volume of a rectangular prism, the same box measured for the space inside rather than the material outside.