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Rectangular Prisms: Volume and Surface Area

Find the volume and surface area of boxes and cubes, and work backwards to find a missing length, width or height.

Practice this skill 25 min Rectangular prisms Leer en español
Key idea

What you'll learn

A rectangular prism is the math name for a box: a cereal box, a shipping carton, a fish tank, a room, a swimming pool, a raised garden bed. In this lesson you will learn to answer the two questions people ask about boxes:

  • How much fits inside? That is volume, measured in cubic units (\text{in}^3, \text{ft}^3, \text{cm}^3).
  • How much material covers the outside? That is surface area, measured in square units (\text{in}^2, \text{ft}^2).

You will also learn to work backwards: if you know the volume (or the surface area) and two of the dimensions, you can find the third one.

Why it matters on the GED: Volume and surface area questions show up on almost every GED Math test, usually as a short word problem with a picture. The formulas are printed on the formula sheet, so the test is really checking whether you can pick the right formula, plug in the right numbers, and keep your units straight. In real life you use the same skills to buy the right amount of soil, paint, concrete, or wrapping paper, or to check whether a couch will fit in a moving truck.

Words to know

  • Rectangular prism — a 3-D shape with 6 flat rectangular faces, like a shoebox.
  • Cube — a rectangular prism whose length, width and height are all equal (a die or a sugar cube).
  • Face — one flat side of the prism. A box has 6 faces that come in 3 matching pairs.
  • Edge — a line where two faces meet. A box has 12 edges.
  • Length (l), width (w), height (h) — the three dimensions of the box (for example 8\text{ in} by 5\text{ in} by 3\text{ in}).
  • Base (B) — the area of the face the box "stands on." For a box with length l and width w on the bottom, B = lw (an 8 \times 5 bottom has B = 40\text{ in}^2).
  • Perimeter of the base (p) — the distance around the base: p = 2l + 2w (for 8 by 5: p = 26\text{ in}).
  • Volume (V) — the space inside, counted in unit cubes (a 2 \times 2 \times 2 cube holds 8 unit cubes).
  • Surface area (SA) — the total area of all 6 faces, as if you unfolded the box flat.
  • Lateral area — the area of the 4 side faces only, without the top and bottom (the label around a box, or the walls of a room).
  • Net — the flat pattern you get when you unfold a box.
  • Cubic units — units for volume: \text{in}^3, \text{ft}^3, \text{yd}^3, \text{cm}^3.
  • Square units — units for area: \text{in}^2, \text{ft}^2, \text{m}^2.
A rectangular prism (box) with its length l, width w and height h labelled; one shaded face and one highlighted edge are pointed out. A box has 6 faces and 12 edges. length l width w height h edge face

All Math words in the glossary

Volume: count the layers of cubes

Rectangular prism. The front bottom edge is the length l, the edge going back is the width w, and the vertical edge is the height h. Hidden edges are dashed. length l width w height h

Picture a box that is 4 units long, 3 units wide and 2 units high.

A box 4 units long, 3 units wide and 2 units high built from unit cubes: the bottom layer has 4 times 3 = 12 cubes and there are 2 layers, so 24 cubes in all. 4 units 3 units 2 layers B = 4 × 3 = 12 cubes per layer; V = 12 × 2 = 24
  1. The bottom layer is a rectangle of 4 \times 3 = 12 unit cubes. That number is the area of the base, B.
  2. The box is 2 layers high, so it holds 12 \times 2 = 24 unit cubes.

That is the whole idea of volume:

V = Bh \qquad \text{(area of the base} \times \text{height)}

For a rectangular prism the base is a rectangle, so B = lw and

V = lwh

Example: a box 8\text{ in} long, 5\text{ in} wide and 3\text{ in} high has V = 8 \times 5 \times 3 = 120\text{ in}^3.

Box 8 inches long, 5 inches wide and 3 inches high. Base area 8 × 5 = 40 square inches; volume 40 × 3 = 120 cubic inches. l = 8 in w = 5 in h = 3 in V = 8 × 5 × 3 = 120 in³

It does not matter which face you call the base. Multiplication can be done in any order: 8 \times 5 \times 3 = 5 \times 3 \times 8 = 120. You will always get the same volume.

Cubes. If every edge is s, then V = s \times s \times s = s^3. A cube with 5\text{ cm} edges has V = 5^3 = 125\text{ cm}^3 (not 15, and not 25).

Cube with every edge 5 centimetres. Volume 5 × 5 × 5 = 125 cubic centimetres. s = 5 cm s = 5 cm s = 5 cm V = 5³ = 125 cm³

Decimals and fractions work the same way. A garden bed 8\text{ ft} by 4\text{ ft} filled 1.5\text{ ft} deep holds 8 \times 4 \times 1.5 = 48\text{ ft}^3 of soil. A box \tfrac{1}{2}\text{ ft} by 2\text{ ft} by 3\text{ ft} has V = \tfrac{1}{2} \times 2 \times 3 = 3\text{ ft}^3.

Raised garden bed 8 feet long and 4 feet wide filled with soil 1.5 feet deep. Volume 8 × 4 × 1.5 = 48 cubic feet. 8 ft 4 ft 1.5 ft V = 8 × 4 × 1.5 = 48 ft³

Units must match. Before you multiply, all three measurements must be in the same unit. A slab 10\text{ ft} by 12\text{ ft} and 6\text{ in} thick: change 6\text{ in} to 0.5\text{ ft} first, so V = 10 \times 12 \times 0.5 = 60\text{ ft}^3.

Concrete slab 12 feet long, 10 feet wide and 6 inches (0.5 foot) thick. Volume 10 × 12 × 0.5 = 60 cubic feet. 12 ft 10 ft 6 in = 0.5 ft V = 10 × 12 × 0.5 = 60 ft³

Volume scales fast. If you double every dimension of a box, the volume is multiplied by 2 \times 2 \times 2 = 8. A 3 \times 4 \times 5 box (60 cubic units) becomes 6 \times 8 \times 10 (480 cubic units).

Two boxes drawn to the same scale: a 3 by 4 by 5 box with volume 60 cubic units, and a 6 by 8 by 10 box with every dimension doubled and volume 480 cubic units, 8 times as much. 5 4 3 10 8 6 V = 60 V = 8 × 60 = 480

Surface area: add up the six faces

Net of a 6 by 4 by 3 rectangular box unfolded into six rectangles: top and bottom are 6 by 4 (24 square units each), front and back are 6 by 3 (18 each), left and right sides are 3 by 4 (12 each). Total 2(24) + 2(18) + 2(12) = 108 square units. back 6×3 top 6×4 front 6×3 3×4 3×4 bottom 6×4

Surface area is the amount of cardboard, paint or wrapping paper needed to cover the outside of the box. Unfold the box and you get a net of six rectangles that come in three matching pairs:

Pair of facesSize of each faceArea of the pair
top and bottoml \times w2lw
front and backl \times h2lh
left and rightw \times h2wh

So one way to write it is SA = 2lw + 2lh + 2wh.

Net of a 6 by 4 by 3 box unfolded flat: top and bottom faces 6 by 4 (24 each), front and back 6 by 3 (18 each), left and right 4 by 3 (12 each). Surface area 108 square units. 4 × 3 = 12 6 × 3 = 18 4 × 3 = 12 6 × 3 = 18 6 × 4 = 24 6 × 4 = 24 6 4 3 SA = 2(24) + 2(18) + 2(12) = 108

The GED formula sheet prints that version for rectangular prisms, and also a second way that works for every right prism:

SA = ph + 2B
  • ph is the lateral area: the four sides unroll into one long rectangle whose width is the perimeter of the base, p, and whose height is h.
  • 2B is the top plus the bottom.
The four side faces of the 6 by 4 by 3 box unroll into one long rectangle 20 units wide (the perimeter 6 + 4 + 6 + 4) and 3 units tall, so the lateral area is ph = 20 × 3 = 60. 6 4 6 4 p = 6 + 4 + 6 + 4 = 20 h = 3 lateral area = ph = 20 × 3 = 60

Example: the 6 \times 4 \times 3 box in the net above. Base B = 6 \times 4 = 24. Perimeter p = 2(6) + 2(4) = 20. Height h = 3.

SA = ph + 2B = 20(3) + 2(24) = 60 + 48 = 108\text{ square units}

Check with the pairs: 2(24) + 2(18) + 2(12) = 48 + 36 + 24 = 108. Same answer.

Cube shortcut: a cube has 6 identical square faces, so SA = 6s^2. A cube with 5\text{ cm} edges has SA = 6 \times 25 = 150\text{ cm}^2.

Read what is being covered. Real objects are not always closed boxes:

  • An open-top box or a swimming pool: only one base, SA = ph + B.
  • The walls of a room: lateral area only, ph. Walls plus ceiling: ph + B.
  • A label that wraps around the sides only: ph.
Three things to cover: a closed box uses all 6 faces (ph + 2B); an open-top box or pool has no top (ph + B); the walls of a room are the 4 sides only (ph). closed: ph + 2B open open top: ph + B open walls only: ph

Working backwards: finding a missing dimension

On the GED you are often given the volume or the surface area and asked for a side. Write the formula, put in everything you know, and solve the equation.

From volume. V = lwh. If you know V, l and w:

h = \frac{V}{lw}

Example: a storage unit holds 960\text{ ft}^3 and its floor is 12\text{ ft} by 10\text{ ft}. The floor area is 120\text{ ft}^2, so h = 960 \div 120 = 8\text{ ft}.

Storage unit with a floor 12 feet by 10 feet and unknown height h. Volume 960 cubic feet, so h = 960 ÷ 120 = 8 feet. 12 ft 10 ft h = ? V = 960 ft³ → h = 960 ÷ 120 = 8 ft

Cube from volume. If a cube has V = 343\text{ in}^3, then s^3 = 343, so s = \sqrt[3]{343} = 7\text{ in} (check: 7 \times 7 \times 7 = 343).

Cube with unknown edge s and volume 343 cubic inches. Since 7 × 7 × 7 = 343, s = 7 inches. s = ? s = ? s = ? V = s³ = 343 in³ → s = 7 in

From surface area. Use SA = ph + 2B, subtract the two bases, then divide by the perimeter.

Example: a closed box with a square base 6\text{ in} on each side has SA = 192\text{ in}^2. Then B = 36, 2B = 72, p = 24:

Closed box with a square base 6 inches by 6 inches and unknown height h; surface area 192 square inches gives h = 5 inches. 6 in 6 in h = ? SA = 192 in² → h = 5 in
192 = 24h + 72 \;\Rightarrow\; 24h = 120 \;\Rightarrow\; h = 5\text{ in}

Always check by putting your answer back into the original formula: 24(5) + 72 = 192 ✓.

Formula

The formulas (on the GED formula sheet)

These appear on the GED formula sheet under right prism:

V = Bh \qquad\qquad SA = ph + 2B
  • B = area of the base. For a rectangle, B = lw (the rectangle area formula A = lw is also on the sheet).
  • h = height of the prism (the edge that is not part of the base).
  • p = perimeter of the base, p = 2l + 2w (also on the sheet).

The sheet also prints the box-only versions under rectangular prism:

V = lwh \qquad SA = 2lw + 2lh + 2wh

Shortcuts you should memorize (they are not printed, but they come straight from the formulas above):

\text{cube: } V = s^3,\; SA = 6s^2

Not on the sheet: unit conversions. Know 12\text{ in} = 1\text{ ft}, 3\text{ ft} = 1\text{ yd}, so 1\text{ yd}^3 = 27\text{ ft}^3 and 1\text{ ft}^3 = 1{,}728\text{ in}^3. Any gallon conversion will be given in the question.

Worked example

Example 1 (basic): volume of a box

A jewelry box is 9\text{ in} long, 4\text{ in} wide and 2\text{ in} high. What is its volume?

Jewelry box 9 inches long, 4 inches wide and 2 inches high; volume 72 cubic inches. 9 in 4 in 2 in
  1. Base area: B = lw = 9 \times 4 = 36\text{ in}^2.
  2. Multiply by the height: V = Bh = 36 \times 2 = 72.

Answer: 72\text{ in}^3

Check: 72 \div (9 \times 4) = 72 \div 36 = 2, the height we started with ✓.

Worked example

Example 2: surface area of a cereal box

A cereal box is 8\text{ in} wide, 3\text{ in} deep and 12\text{ in} tall. How much cardboard covers the outside?

Cereal box 8 inches wide, 3 inches deep and 12 inches tall, standing on its 8 by 3 base. 8 in 3 in 12 in
  1. The box stands on an 8 \times 3 base: B = 8 \times 3 = 24\text{ in}^2.
  2. Perimeter of the base: p = 2(8) + 2(3) = 16 + 6 = 22\text{ in}.
  3. Lateral area: ph = 22 \times 12 = 264\text{ in}^2.
  4. Two bases: 2B = 2 \times 24 = 48\text{ in}^2.
  5. SA = 264 + 48 = 312.
Net of the cereal box 8 by 3 by 12 inches: front and back 8 by 12 (96 each), sides 3 by 12 (36 each), top and bottom 8 by 3 (24 each). Total 312 square inches. 3 × 12 = 36 8 × 12 = 96 3 × 12 = 36 8 × 12 = 96 8 × 3 = 24 8 × 3 = 24 8 in 3 in 12 in SA = 2(96) + 2(36) + 2(24) = 312 in²

Answer: 312\text{ in}^2

Check: pairs of faces: 2(8 \cdot 3) + 2(8 \cdot 12) + 2(3 \cdot 12) = 48 + 192 + 72 = 312 ✓.

Worked example

Example 3 (real world): ordering concrete

A patio slab will be 20\text{ ft} long, 10\text{ ft} wide and 6\text{ in} thick. Concrete is sold by the cubic yard (1\text{ yd}^3 = 27\text{ ft}^3), in whole cubic yards only. How many cubic yards should be ordered?

Patio slab 20 feet long, 10 feet wide and 6 inches (0.5 foot) thick; volume 100 cubic feet, about 3.7 cubic yards. 20 ft 10 ft 6 in = 0.5 ft V = 20 × 10 × 0.5 = 100 ft³
  1. Make the units match: 6\text{ in} = 6 \div 12 = 0.5\text{ ft}.
  2. Volume in cubic feet: V = 20 \times 10 \times 0.5 = 100\text{ ft}^3.
  3. Convert to cubic yards: 100 \div 27 \approx 3.70\text{ yd}^3.
  4. Concrete comes in whole yards and 3 would not be enough, so round up.

Answer: order 4 cubic yards.

Check: 4 \times 27 = 108\text{ ft}^3 \ge 100\text{ ft}^3 ✓, but 3 \times 27 = 81 < 100, so 3 yards would run short.

Worked example

Example 4 (work backwards): missing height

A shipping box has a volume of 1{,}152\text{ in}^3. It is 16\text{ in} long and 9\text{ in} wide. How tall is it?

Shipping box 16 inches long and 9 inches wide with unknown height h; volume 1,152 cubic inches, so h = 8 inches. 16 in 9 in h = ? V = 1,152 in³ = 144 × h
  1. Write the formula: V = lwh.
  2. Substitute: 1{,}152 = 16 \times 9 \times h = 144h.
  3. Divide both sides by 144: h = 1{,}152 \div 144 = 8.

Answer: 8\text{ in}

Check: 16 \times 9 \times 8 = 144 \times 8 = 1{,}152 ✓.

Worked example

Example 5 (GED-hard): height from surface area

A closed gift box has a square base 5\text{ cm} on each side. It takes exactly 210\text{ cm}^2 of paper to cover it with no overlap. What is the height of the box?

Closed gift box with a square base 5 centimetres on each side and unknown height h; 210 square centimetres of paper covers it. 5 cm 5 cm h = ? SA = 210 cm²
  1. Base area: B = 5 \times 5 = 25\text{ cm}^2, so 2B = 50\text{ cm}^2.
  2. Perimeter of the base: p = 4 \times 5 = 20\text{ cm}.
  3. Use SA = ph + 2B: 210 = 20h + 50.
  4. Subtract 50: 20h = 160.
  5. Divide by 20: h = 8.

Answer: 8\text{ cm}

Check: 2(5 \cdot 5) + 2(5 \cdot 8) + 2(5 \cdot 8) = 50 + 80 + 80 = 210 ✓.

Calculator tip

On the TI-30XS MultiView

  • Volume: just multiply: 8 × 5 × 3 enter → 120.
  • Cube of a number: use the ^ key: 5 ^ 3 enter → 125. (The x² key only squares, which gives 25, the area of one face.)
  • Cube root (side of a cube from its volume): type the index 3, then 2nd ^ (\sqrt[n]{\;}), then 343, enter → 7.
  • Surface area in one line: use parentheses: 2 × ( 8 × 3 + 8 × 12 + 3 × 12 ) enter → 312.
  • Fractions: the n/d key enters a fraction like \tfrac{1}{2}; the toggle key switches the answer between fraction and decimal.

When not to bother: small whole-number boxes (2 \times 3 \times 4) are faster in your head. Use the calculator for decimals, big numbers, and unit conversions like \div 27 or \div 1{,}728.

Common mistake

Common mistakes

  • Adding instead of multiplying. ✗ 8 + 5 + 3 = 16. ✓ Volume multiplies: 8 \times 5 \times 3 = 120.
  • Mixing up volume and surface area. Volume (fill, hold, contain, cubic units) is lwh. Surface area (cover, paint, wrap, square units) is ph + 2B. Read the verb in the question.
  • Forgetting to double. ✗ lw + lh + wh counts only 3 of the 6 faces. ✓ Every face has a twin: 2lw + 2lh + 2wh.
  • Covering faces that aren't there. A pool has no top; room walls have no floor or ceiling. ✗ Using the full ph + 2B. ✓ Include only the faces the problem describes.
  • Mismatched units. ✗ 10\text{ ft} \times 12\text{ ft} \times 6\text{ in} = 720. ✓ Change 6\text{ in} to 0.5\text{ ft} first: 60\text{ ft}^3.
  • Cube confusion. ✗ 5^3 = 15 or 5^3 = 25. ✓ 5^3 = 5 \times 5 \times 5 = 125.
Test strategy

GED test strategy

  • Read the question twice. Underline what is asked: volume (fill, hold, capacity) or surface area (cover, paint, wrap, material)? Is the box open or closed?
  • Write the formula first, then substitute. Copying V = Bh or SA = ph + 2B from the formula sheet keeps you from mixing them up.
  • Check the units in the answer choices. If the answer choices are in \text{ft}^3 but a measure is in inches, convert first.
  • Use the wrong answers as clues. GED distractors come from common errors. If you see both 120 and 158, one is the volume and one is the surface area of the same box. Make sure you picked the one the question asked for.
  • Estimate. 11.8 \times 6.1 \times 3.9 \approx 12 \times 6 \times 4 = 288. Choose the answer close to that.
  • Work-backwards questions: plug each answer choice into the formula and see which one gives the stated volume. That is often faster than solving.
  • Rounding up for real purchases. If you need 3.7 bags or 1.67 gallons, you must buy 4 bags or 2 gallons.

Check yourself

Rectangular prisms

What is the volume of the box shown?

Rectangular box 8 inches long, 5 inches wide, and 3 inches high. 8 in 5 in 3 in
Answer choices
Hint

Volume of a rectangular prism: V = Bh, where B is the area of the base.

Check yourself

Rectangular prismsFill in

A closed gift box has the dimensions shown.

Closed gift box shaped like a rectangular prism: length 10 inches, width 6 inches, height 4 inches. 10 in 6 in 4 in

How many square inches of wrapping paper are needed to cover the box exactly, with no overlap?

\text{in}^2

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Wrapping paper covers all 6 faces. Use SA = ph + 2B (formula sheet), where p is the perimeter of the base.

Summary

  • Volume = space inside: V = Bh = lwh, in cubic units. A cube: V = s^3.
  • Surface area = total area of the 6 faces: SA = ph + 2B = 2lw + 2lh + 2wh, in square units. A cube: SA = 6s^2.
  • V = lwh, SA = 2lw + 2lh + 2wh, V = Bh and SA = ph + 2B are all on the GED formula sheet; know what B, p and h mean.
  • Make all units match before you calculate; open boxes and walls use fewer faces.
  • To find a missing dimension, substitute what you know and solve: h = \dfrac{V}{lw}, or h = \dfrac{SA - 2B}{p}.
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