Lesson 7 of 8 · Grades 3 to 6
Volume of rectangular prisms
A rectangular prism is a box with six flat rectangular faces. Its volume is the space inside, and it is found by multiplying three numbers: length, width and height.
The volume rule
A rectangular prism is a solid box with six flat rectangular faces. Bricks, cereal boxes and fish tanks are all rectangular prisms. Its volume is the space inside. Counting unit cubes one at a time is slow once a box gets big, so we use a rule instead.
The prism below is 5 cm long, 3 cm wide and 4 cm tall. The bottom layer holds 5 × 3 = 15 cubes. There are 4 layers, so the volume is 15 × 4 = 60 cubic centimetres. The same work is written 5 × 3 × 4 = 60 cm³.
The rule in three numbers
The layer method hides a simple rule: volume of a rectangular prism = length × width × height. All three lengths must use the same unit before you multiply. The answer is in cubic units, because three lengths were multiplied together.
Multiplication can be done in any order, so it does not matter which face you call the base. The same box is 15 cubes in a layer with 4 layers, or 12 cubes in a layer with 5 layers. Both give 60 cubes. The names length, width and height are just labels for the three directions.
A cube is a special prism
A cube is a rectangular prism whose three edge lengths are equal. Its volume rule becomes edge × edge × edge, or side cubed. A cube with an edge of 4 cm has a volume of 4 × 4 × 4 = 64 cm³. A cube with an edge of 6 cm has a volume of 216 cm³.
Cubes are the unit blocks of volume. A cubic centimetre is a cube with an edge of 1 cm, and a cubic metre is a cube with an edge of 1 m. A volume of 24 cm³ means the space could be filled by 24 such cubes with no gaps.
Finding a missing edge
When the volume and two edges are known, division finds the third edge. Multiply the two edges you know, then divide the volume by that number. A prism with a volume of 72 cm³, a length of 6 cm and a width of 4 cm is 72 ÷ (6 × 4) = 72 ÷ 24 = 3 cm tall.
Check the answer by putting it back into the rule: 6 × 4 × 3 = 72, which is the volume we started with. This check catches a slip with the division or the units.
Worked example
A fish tank is 40 cm long, 20 cm wide and 30 cm tall. Find its volume.
Step 1: find the area of the base. The base is 40 cm by 20 cm, so 40 × 20 = 800 cm².
Step 2: multiply the base area by the height. 800 × 30 = 24,000.
Step 3: check with the three numbers in another order. 40 × 20 × 30 also gives 24,000.
Step 4: label the unit. Three lengths in centimetres give cubic centimetres, so the volume is 24,000 cm³.
Answer: 24,000 cm³.
Watch out for this
Volume multiplies three numbers, so it is easy to lose one. A prism 5 cm by 3 cm by 4 cm has a volume of 60 cm³, not 12 cm² and not 12 cm³. Length × width alone gives the area of a face, and a face is flat, not a solid.
Remember
- Volume of a rectangular prism = length × width × height.
- Use the same unit for all three lengths.
- Write the answer in cubic units, such as cm³ or m³.
- To find a missing edge, divide the volume by the other two edges multiplied.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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A prism is 6 cm long, 3 cm wide and 2 cm tall. What is its volume?
Show answer
36 cm³. 6 × 3 × 2 = 36.
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A box is 10 m long, 5 m wide and 2 m tall. What is its volume?
Show answer
100 m³. 10 × 5 × 2 = 100.
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A prism has a volume of 24 cm³, a length of 4 cm and a width of 3 cm. How tall is it?
Show answer
2 cm. 4 × 3 = 12, and 24 ÷ 12 = 2.
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A cube has edges of 4 cm. What is its volume?
Show answer
64 cm³. 4 × 4 × 4 = 64.
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A tank is 20 cm long, 10 cm wide and 10 cm tall. What is its volume?
Show answer
2,000 cm³. 20 × 10 × 10 = 2,000.
Puzzle · Box volume
Puzzle time
A cereal box is 20 cm long, 10 cm wide and 25 cm tall. What is its volume?
Show a hint
Multiply the three edge lengths: length × width × height.
Show the answer
Answer: 5,000 cm³. 20 × 10 = 200, and 200 × 25 = 5,000, so the volume is 5,000 cm³.