Lesson 12: Volume of Rectangular Prisms

How much water fits in a fish tank? How many boxes can a delivery van carry? Questions about filling a three-dimensional space are answered by volume. Where area counts unit squares on a flat surface, volume counts unit cubes packed into a solid. In this lesson you will find the volume of rectangular prisms and cubes, work backwards to find a missing dimension, and convert your answers into practical units like litres.

Counting cubes to build the formula

Imagine a box 5 cubes long, 4 cubes wide and 3 cubes tall. The bottom layer holds 5 × 4 = 20 cubes, and there are 3 layers, so the box holds 20 × 3 = 60 unit cubes. That layering idea is the formula: Volume = length × width × height, or V = l × w × h. For a cube with edge s, every dimension is the same: V = s³. Volume is measured in cubic units: cm³, m³, in³.

Worked example 1. A box measures 5 cm × 4 cm × 3 cm. V = 5 × 4 × 3 = 60 cm³.

Worked example 2. A cube has edge 4 m. V = 4³ = 4 × 4 × 4 = 64 m³. Another cube, edge 5 in, holds 5³ = 125 in³ — more than double the first cube's cousin of edge 4 (64), because volume grows with the cube of the edge. Doubling an edge multiplies volume by 8, a fact that surprises everyone the first time.

Volume in the real world: litres in a tank

A fish tank measures 60 cm × 30 cm × 40 cm. Its volume is 60 × 30 × 40 = 72,000 cm³. Since 1 litre = 1,000 cm³, the tank holds 72,000 ÷ 1,000 = 72 litres of water (before gravel and fish take their share). Shops sell tanks by litres, movers price lorries by cubic metres, and bakers choose tins by volume — being able to convert between the formula and the real unit is the skill that makes volume useful.

Worked example 3 (fractional edges). A planter measures 2.5 m × 4 m × 2 m. V = 2.5 × 4 × 2 = 20 m³ of soil. Decimals multiply just like whole numbers; do 25 × 4 × 2 = 200 first, then place the decimal point.

Finding a missing dimension

If you know the volume and two dimensions, divide to find the third.

Worked example 4. A prism has volume 96 cm³, length 8 cm and width 4 cm. The base area is 8 × 4 = 32 cm², so the height is 96 ÷ 32 = 3 cm. Check: 8 × 4 × 3 = 96 ✓.

Comparing volumes

Which holds more: crate A at 4 × 4 × 4, or crate B at 8 × 3 × 2? A: 64. B: 48. The squat, cube-like crate A wins despite B's longer side — long and thin loses to compact and cube-ish. This is why suitcases and storage boxes tend towards chunky shapes.

Common mistakes

• Using surface-area thinking. Volume multiplies three dimensions. 5 + 4 + 3 = 12 is not the volume of the 5 × 4 × 3 box; 5 × 4 × 3 = 60 is.
• Forgetting the third dimension. Multiplying only length × width gives the area of the base (20 cm²), not the volume (60 cm³).
• Unit slips. Answers in cm² or cm are impossible for volume — it must be cubic units, cm³. And 72,000 cm³ is 72 litres, not 72,000 litres.
• Assuming doubling doubles. Doubling every edge of a cube multiplies its volume by 2³ = 8, not 2.

Exam Experience

1. What is the volume of a rectangular prism measuring 6 cm × 3 cm × 2 cm? A) 11 cm³ B) 22 cm³ C) 36 cm³ D) 66 cm³

2. What is the volume of a cube with edge 5 cm? A) 15 cm³ B) 25 cm³ C) 75 cm³ D) 125 cm³

3. Crate A measures 4 × 4 × 4. Crate B measures 8 × 3 × 2. Which crate has the greater volume, and by how much? A) B, by 16 B) A, by 16 C) They are equal D) A, by 8

4. (Constructed response) A storage bin measures 30 in long, 18 in wide and 12 in tall. (a) Find the volume of the bin. (b) The bin is filled with small cubes of edge 6 in. How many cubes fit? (c) Show a second way to answer part (b) by counting cubes along each edge, and confirm both methods agree.

Answer key and explanations

• 1. C — 6 × 3 × 2 = 36 cm³.
• 2. D — 5³ = 5 × 5 × 5 = 125 cm³.
• 3. B — A: 64; B: 48. A is greater by 64 − 48 = 16.
• 4. (a) 30 × 18 × 12 = 6,480 in³. (b) Each small cube has volume 6³ = 216 in³, so 6,480 ÷ 216 = 30 cubes. (c) Along the edges: 30 ÷ 6 = 5 cubes long, 18 ÷ 6 = 3 wide, 12 ÷ 6 = 2 tall, so 5 × 3 × 2 = 30 cubes — both methods agree.

Scoring guide for question 4 (3 points)

• 1 point for the volume, 6,480 in³, from multiplying all three dimensions.
• 1 point for 30 cubes via the volume method (6,480 ÷ 216).
• 1 point for 30 cubes via the edge-counting method (5 × 3 × 2), with an explicit statement that the methods agree.

Going further: choosing and converting

Worked example 5. Which aquarium holds more water? Tank A: 50 cm × 25 cm × 30 cm = 37,500 cm³ = 37.5 litres. Tank B: 40 cm × 40 cm × 25 cm = 40,000 cm³ = 40 litres. Tank B holds more, even though Tank A is longer — volume rewards balanced dimensions, a fact worth remembering when a shop tries to sell you the long, thin, impressive-looking tank.

Worked example 6. A removal van has a cargo space of 3 m × 2 m × 2 m = 12 m³. Boxes of 0.5 m³ each: 12 ÷ 0.5 = 24 boxes. Dividing a big volume by a small one to count how many fit is one of the most common volume calculations in working life.

Worked example 7. A raised garden bed measures 1.5 m × 2 m × 0.5 m. V = 1.5 × 2 × 0.5 = 1.5 m³ of compost. Multiplying by a half at the end is painless: halve any one factor (2 → 1) and the rest follows.

Worked example 8. A prism has volume 240 cm³ and a base of 10 cm × 6 cm. Base area = 60 cm², so the height is 240 ÷ 60 = 4 cm. The same divide-by-the-base move solves every "missing dimension" problem, whatever the shape of the numbers.

Quick practice

• 1. Volume of a prism 7 cm × 2 cm × 3 cm.
• 2. Volume of a cube with edge 6 cm.
• 3. Convert 500 cm³ into litres.
• 4. A box has volume 180 cm³, length 9 cm and width 5 cm. Find its height.

Answers. 1. 7 × 2 × 3 = 42 cm³. 2. 6³ = 216 cm³. 3. 500 ÷ 1,000 = 0.5 litres. 4. Base = 45 cm², height = 180 ÷ 45 = 4 cm.

Investigation: the rain barrel and the book box

A rectangular rain barrel measures 80 cm × 50 cm × 60 cm. Its volume is 80 × 50 × 60 = 240,000 cm³, and since 1 litre = 1,000 cm³, that is 240 litres of stored water. A tap fills it at 12 litres per minute, so a full barrel takes 240 ÷ 12 = 20 minutes, and a half-full barrel holds 120 litres — volume scales exactly with the height of the water, because the base never changes. Now pack it (figuratively): a storage box measures 40 cm × 30 cm × 20 cm, volume 24,000 cm³. A paperback book measures 20 cm × 15 cm × 4 cm, volume 1,200 cm³. By volume, 24,000 ÷ 1,200 = 20 books fit. Happily, the dimensions divide exactly too (2 along the length, 2 along the width, 5 along the height: 2 × 2 × 5 = 20), so the volume answer is genuinely achievable. That double-check matters: volume division gives an upper bound, and only tidy dimensions guarantee the bound can be reached. Real packing — vans, suitcases, shipping containers — lives in the gap between the two methods.

Challenge corner

Pour 30 litres of water into an empty aquarium whose base is 40 cm × 20 cm. How deep is the water? First convert: 30 litres = 30,000 cm³. The base area is 800 cm², so the depth is 30,000 ÷ 800 = 37.5 cm. This is the missing-dimension skill wearing a useful disguise — and a genuinely practical one, since fishkeepers dose medicine by water depth and volume. A companion check on volume addition: two boxes of 4 × 3 × 2 (volume 24 each) joined end to end make a solid of 8 × 3 × 2, whose volume is 48 = 24 + 24 ✓. Volumes always add when solids join. (Surface areas, as you saw in the last lesson, do not — the joined faces disappear. Volume is the honest measure; surface is the shy one.) Together these two facts settle most "combined solid" problems you will meet this year.