How to find the surface area of a box, a prism or a cylinder

Your child has a cereal box on the table and a worksheet asking for its surface area, and the answer key says 62 while their answer says 30. That mix-up between the outside of a box and the space inside it is the most common one in grade 6 geometry. Learning how to find surface area takes one idea: unfold the shape into flat pieces, find the area of each piece and add them up.

What is surface area?

Surface area is the total area of all the faces of a 3D shape. OpenStax Prealgebra 2e, section 9.6 puts it in terms of a painted crate: "The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides." Wrapping paper, paint and the cardboard in a box are all surface area questions.

The tool that makes surface area visible is the net, the flat pattern you get when you cut a box along its edges and lay it out. Common Core standard 6.G.A.4 asks sixth graders to "represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures." In grade 7, standard 7.G.B.6 widens the job to "area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms."

Surface area builds on the area of a rectangle, length × width. If that feels shaky, the guide to area and perimeter covers it first.

Why surface area uses square units, not cubic

Surface area is measured in square units because every face is a flat area, one length times another. A face 5 inches by 3 inches covers 15 square inches, written 15 in². Volume multiplies three lengths, so it comes out in cubic units such as in³.

The NIST page on SI units shows the same pattern for the metric system: area is m² and volume is m³. The little 2 and 3 count how many lengths were multiplied. When your child writes "62 cubic units" for a surface area, the unit itself shows the mix-up.

How to find the surface area of a rectangular prism, step by step

The surface area of a rectangular prism is the sum of its 6 rectangular faces, which come in 3 matching pairs. Take a box 5 units long, 3 units wide and 2 units high.

1. Name the three edges. Write l = 5, w = 3 and h = 2. Every face of the box uses two of those three numbers.
2. Draw the net. Sketch the six rectangles unfolded, as in the box diagram, and write the size on each one: two 5 × 3, two 5 × 2 and two 3 × 2.
3. Find the area of one face from each pair. Top: 5 × 3 = 15. Front: 5 × 2 = 10. Side: 3 × 2 = 6.
4. Add the three areas. 15 + 10 + 6 = 31. That covers half the box, one face from each pair.
5. Double the sum. 2 × 31 = 62. The formula 2(lw + lh + wh) does steps 3 to 5 in one line: 2(15 + 10 + 6) = 62.
6. Write the unit and check. The answer is 62 square units. Add all six faces one by one as a check: 15 + 15 + 10 + 10 + 6 + 6 = 62.

The formula 2(lw + lh + wh) and the six-face sum give the same answer because of the distributive property, which the guide to properties of multiplication explains. OpenStax writes it as S = 2LH + 2LW + 2WH, and its own worked example, a 14 × 9 × 17 cm solid, comes to 1,034 square centimeters.

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The part people get wrong: counting only three faces

Counting only the three faces you can see is the most common surface area mistake. OpenStax points this out directly: of a box's faces, "You can see three of them," and "for each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all."

The second mistake is multiplying the three edges, 5 × 3 × 2 = 30, which is the volume of the box. Volume has its own guide, how to find volume; for surface area, the three edges are always multiplied two at a time.

Surface area formulas for each shape, side by side

Surface area formulas all come from the same net idea: list the faces, find each area and add. The examples use whole numbers, and π ≈ 3.14 where a circle appears.

The cube formula 6s² uses an exponent: s² means s × s, so a cube with 4-unit edges has 6 faces of 16 square units each. The guide to exponents covers the notation.

How to find the surface area of a triangular prism

The surface area of a triangular prism is the two triangle ends plus the three rectangles that join them. Take a prism whose ends are right triangles with sides 3, 4 and 5, and whose length is 10.

Each triangle is ½ × 3 × 4 = 6, so the two ends make 12. The three rectangles are 3 × 10 = 30, 4 × 10 = 40 and 5 × 10 = 50, which make 120. The total is 12 + 120 = 132 square units.

The three rectangles share the length 10, so you can also multiply the triangle's perimeter by the length: (3 + 4 + 5) × 10 = 120. Cylinders and spheres fall outside 6.G.A.4 and 7.G.B.6, but OpenStax teaches the cylinder in the same section as boxes, and many middle school textbooks do too.

Kitchen-table activities for surface area

Surface area makes the most sense when your child unfolds a real object. Four activities need nothing more than the recycling bin, a pencil and a ruler.

Unfold a cereal box. Open the glued flaps, lay the box flat and you have its net. Measure the length, width and height to the nearest inch, label each panel with its area and add them. Then compare with 2(lw + lh + wh); the flaps that overlap at the glue line are extra cardboard and are not part of the surface area.

Wrap a gift. A box 12 × 8 × 4 inches has faces of 96, 48 and 32 square inches, so its surface area is 2(96 + 48 + 32) = 352 square inches. Cut a piece of paper a little bigger than that for the overlap, and ask your child why the paper must be larger than the exact answer.

Peel a can label. OpenStax describes this one: "If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle." The label's width is the circumference of the can, 2πr, and its height is the height of the can. The guide to circumference and area of a circle covers 2πr and πr².

The worked cylinder uses r = 3 and h = 10 and keeps π until the last step:

• Two circles: 2 × π × 3² = 2 × π × 9 = 18π.
• The rectangle: 2 × π × 3 × 10 = 60π.
• Total: 18π + 60π = 78π. With π ≈ 3.14, 78 × 3.14 = 244.92 square units.

Count the paint for a room. A bedroom 14 feet long, 12 feet wide and 8 feet high has two walls of 14 × 8 = 112 square feet and two of 12 × 8 = 96, so the walls total 2 × 112 + 2 × 96 = 416 square feet. Subtract the door and window, then compare with the coverage printed on the paint can.

Free printable worksheetMultiplication — 6th GradersWorksheet · PDF

The rule: draw the net first

Drawing the net before any formula is the habit that prevents most errors. A net shows every face once, so a missing face or a doubled one is easy to spot. Once your child can draw the net of a box, a prism and a pyramid without help, the formulas become shortcuts for something they can already see.

How to check progress with surface area

Surface area progress shows up in five quick tasks. Give one task a day for a week and watch which one is still slow.

• Box: find the surface area of a 4 × 3 × 2 box. The answer is 2(12 + 8 + 6) = 52 square units.
• Cube: find the surface area of a cube with 5-unit edges. The answer is 6 × 25 = 150 square units.
• Net: draw the net of a triangular prism. It must show 2 triangles and 3 rectangles.
• Units: say the unit for 52 and for the volume of the same box, 24. Square units for the first, cubic units for the second.
• Real object: measure a tissue box and find its surface area to the nearest square inch.

A five-minute test

A five-minute surface area test is one box, one cube and one prism, with the net drawn for each. Score the net and the arithmetic separately.

A correct net with a wrong total points to the multiplication facts: a 6th grader computes 7 × 8 and 6 × 9 dozens of times in one worksheet. The math game for 6th graders and the times tables game keep those facts quick. A wrong net points back to the unfolding activities above.

Common surface area mistakes and edge cases

Five mistakes cause most wrong surface area answers, and each has a short fix.

• Three faces instead of six: the answer comes out half the size. Double the sum of the three different faces, or list all six.
• Multiplying all three edges: 5 × 3 × 2 = 30 is the volume. Multiply the edges two at a time.
• Cubic units: writing cm³ for a surface area. Areas always take square units.
• Mixed units: a box 2 feet × 6 inches × 6 inches must be converted first, to 24 × 6 × 6 inches, giving 2(144 + 144 + 36) = 648 square inches.
• Diameter for radius: a can 6 units across has r = 3. Halve the diameter before you use 2πr² + 2πrh.

Edge case: doubling every edge multiplies surface area by 4

Doubling every edge of a shape multiplies its surface area by 4. A 5 × 3 × 2 box has a surface area of 62, and a 10 × 6 × 4 box has 2(60 + 40 + 24) = 248, which is 4 × 62. Each face is a length times a length, and both lengths doubled, so each face area is 2 × 2 = 4 times bigger.

Volume grows faster still. The small box holds 30 cubic units and the big one holds 240, which is 8 times as much, because volume multiplies three doubled lengths: 2 × 2 × 2 = 8. The volume guide has more examples of that scaling.

Your surface area checklist and next steps

• Net first: sketch the flat pattern and count the faces before you calculate.
• Two at a time: each face uses two edges, never all three.
• Pairs: for a box, find 3 face areas, add them and double the sum.
• Circles: halve a diameter, keep π until the last step and say which value of π you used.
• Units: convert to one unit first and write the answer in square units.
• Check: add every face of the net one by one and compare with the formula.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and short daily practice that keeps the multiplication behind every face area quick. Your child can also challenge 1 to 4 friends in real-time math battles.