Understanding and measuring volume
Volume as packed unit cubes, counting cubic units, and the length x width x height formula
About four lessons of 45 to 60 minutes
How much does a box actually hold?
A shoe box and a shoe-box-shaped cereal box might look about the same size from the front, but which one holds more? Area tells you how much flat space a shape covers, but a box is not flat -- it has a third dimension, depth, and the amount of space inside it is called volume.
Just as area is measured by counting how many unit squares tile a flat shape, volume is measured by counting how many unit cubes pack solidly inside a solid figure, with no gaps and no overlaps. Today you will build that picture with unit cubes, then find a shortcut formula so you never have to count one-by-one again.
- A single unit cube1 unit long, 1 unit wide, 1 unit tall: the building block of volume
- A box 3 x 2 x 4 unit cubes24 unit cubes fit inside with no gaps: the volume is 24 cubic units
- A fish tank 40 cm x 20 cm x 25 cm20,000 cubic centimetres of water fit inside
- The shortcut formulaV = length x width x height, the same answer counting cubes gives, much faster
What students will be able to do
Students will understand volume as the number of unit cubes that pack a solid figure without gaps or overlaps, measure volume by counting unit cubes for figures made of cubic centimetres, cubic inches, or cubic feet, and find the volume of a right rectangular prism both by counting unit cubes and by multiplying its three edge lengths.
- I can explain that volume is the number of unit cubes that exactly pack a solid, with no gaps.
- I can count layers of unit cubes to find the volume of a prism.
- I can find volume using V = length x width x height.
- I can explain why counting layers and multiplying edge lengths give the same answer.
- I can solve a real-world problem involving the volume of a box or container.
Standards this unit teaches
- 5.MD.C.3Common Core (US)Understand volume
Understand volume as the amount of unit cubes that fit inside a solid figure without gaps.
- 5.MD.C.4Common Core (US)Measure volume by counting
Measure the volume of solid figures by counting unit cubes in cubic centimetres, inches, and feet.
- 5.MD.C.5Common Core (US)Volume of rectangular prisms
Find the volume of right rectangular prisms by counting unit cubes and by multiplying edge lengths.
- AC9M5SP01Australian Curriculum v9 (ACARA)Cross-sections and prisms (Year 5)
Compare the parallel cross-sections of objects and recognise how they relate to right prisms. This Year 5 descriptor builds the same stacked-layers picture of a prism that volume by counting unit cubes rests on.
- AC9M7M02Australian Curriculum v9 (ACARA)Volume of prisms (Year 7)
Use formulas and suitable units to solve problems about the volume of right prisms, including rectangular and triangular prisms. Australia's formal volume-formula descriptor sits at Year 7, so this US Grade 5 unit runs about two years ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Volume
- the amount of space inside a solid figure, measured in cubic units
- Unit cube
- a cube 1 unit long, 1 unit wide, and 1 unit tall, the building block used to measure volume
- Cubic unit
- the volume of one unit cube, written as a small 3 (such as cubic centimetres, cm to the power 3)
- Rectangular prism
- a solid box shape with 6 rectangular faces, like a shoe box or a brick
- Edge length
- the length of one side of a solid figure; a rectangular prism has three different edge lengths (length, width, height)
- Layer
- one flat slice of unit cubes stacked to build up a solid figure
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Volume is packed unit cubes
ConcreteGive each group a small pile of centimetre cubes and a small open box. Pack the cubes into the box so they fit with no gaps and no overlaps, and count them. That count is the volume of the box, measured in cubic units, because each unit cube takes up exactly 1 cubic unit of space.
Volume for a solid is the 3D version of area for a flat shape. Area counts unit squares tiling a surface; volume counts unit cubes packing a solid all the way through, not just covering its outside.
- What is a unit cube, and how much space does one take up?
- How is measuring volume like measuring area, and how is it different?
2. Counting volume by layers
PictorialCounting every single cube one at a time is slow for a big box, so count by layers instead. Build a box 3 cubes long, 2 cubes wide, and 4 cubes tall. Each flat layer has 3 x 2 = 6 cubes. There are 4 layers stacked up, so the total is 6 x 4 = 24 cubes.
This layer method is the bridge between physically counting and the formula that comes next: the cubes in one layer is length x width, and the number of layers is the height, so the total is (length x width) x height.
A box is 5 cm long, 3 cm wide, and 2 cm tall, built from centimetre cubes. Find its volume by counting layers.
- One layer (the base) has 5 x 3 = 15 cubes.
- There are 2 layers stacked to the given height.
- Total cubes: 15 x 2 = 30.
Answer: The box holds 30 cubic centimetres.
- How many cubes are in one layer of a box 5 cm by 3 cm?
- Why do you multiply the cubes in one layer by the number of layers?
3. The formula: V = length x width x height
AbstractThe layer method always gives the same result as multiplying all three edge lengths together, because 'cubes in one layer' is length x width and 'number of layers' is the height. So the formula V = length x width x height is just a shortcut for the exact same counting, and it works for any right rectangular prism, however big.
Use the formula on a real container: a fish tank measuring 40 cm long, 20 cm wide, and 25 cm tall. V = 40 x 20 x 25. Multiply in steps: 40 x 20 = 800, then 800 x 25 = 20,000. The tank holds 20,000 cubic centimetres, no counting of 20,000 individual cubes required.
A storage box measures 10 cm by 8 cm by 15 cm. Find its volume using the formula.
- V = length x width x height = 10 x 8 x 15.
- 10 x 8 = 80.
- 80 x 15 = 1,200.
Answer: The box holds 1,200 cubic centimetres.
- Why does multiplying length x width x height give the same answer as counting layers?
- A cube has all edges 3 units long. What is its volume?
Common misconceptions and how to address them
MisconceptionVolume and surface area are the same thing, both counted from the outside of the shape.
Why it happens: Both are 'how big is this box' measurements, so students conflate the space inside with the flat faces on the outside.
How to address it: Volume is the space packed inside a solid (measured in cubic units, cm to the power 3); surface area is the flat area of all the outer faces added up (measured in square units, cm to the power 2). Physically packing cubes inside a box, versus wrapping paper around its outside, makes the difference concrete.
MisconceptionYou can find volume by adding the three edge lengths, since 'you use all three numbers'.
Why it happens: Students over-generalise from perimeter, where you add side lengths, without noticing that volume is a 3D multiplying relationship, not an additive one.
How to address it: Show the layer count directly: a 3 x 2 x 4 box has 24 packed cubes, not 3 + 2 + 4 = 9. Counting cubes by hand for a small example, then comparing to both the sum and the product, makes the correct operation obvious.
MisconceptionThe order you multiply the three edge lengths in changes the volume.
Why it happens: Students are unsure whether length, width, and height must be multiplied in a fixed order.
How to address it: Multiplication can be done in any order and gives the same product, so 3 x 2 x 4, 4 x 3 x 2, and 2 x 4 x 3 all equal 24. It does not matter which edge you call length, width, or height, as long as you multiply all three.
Guided practice (with answers)
1. A prism is built from unit cubes: 4 long, 3 wide, 2 tall. What is its volume?
Answer: 24 cubic units. One layer is 4 x 3 = 12 cubes; 2 layers gives 12 x 2 = 24.
2. A box measures 6 inches by 2 inches by 5 inches. Find its volume.
Answer: 60 cubic inches, since 6 x 2 x 5 = 60.
3. A cube has every edge 3 units long. What is its volume?
Answer: 27 cubic units, since 3 x 3 x 3 = 27.
4. A storage box measures 10 cm by 8 cm by 15 cm. Find its volume.
Answer: 1,200 cubic centimetres, since 10 x 8 x 15 = 1,200.
5. How many unit cubes are needed to completely fill a box 3 units by 3 units by 3 units?
Answer: 27 unit cubes, since 3 x 3 x 3 = 27.
Independent practice worksheets
Set the matching ChalkBee volume worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with counting layers of unit cubes before moving straight to the formula.
Differentiation
- Use real interlocking cubes so students physically build and count small prisms before moving to drawings or the formula.
- Draw prisms as clearly labelled layers (with the cubes-per-layer count written on each layer) before jumping to the abstract formula.
- Keep early formula practice to whole-number edge lengths under 10 so the multiplication itself is not the bottleneck.
- Provide a written formula card (V = length x width x height) for students to refer to while the layer meaning is still being secured.
- Find the missing edge length given the volume and the other two edges (a first taste of solving for an unknown).
- Compare the volumes of two differently shaped boxes with the same total surface area, or the same volume with different shapes, to see they are independent measurements.
- Solve multi-step problems, such as how many identical boxes of a given volume fit inside a larger shipping container.
- Preview Grade 6 by exploring what happens to the formula when an edge length is a fraction, such as 2 and 1/2 units.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling counting, the formula, and a real context.
1. A prism is 6 units long, 4 units wide, and 3 units tall. Find its volume.
Answer: 72 cubic units, since 6 x 4 x 3 = 72.
2. A box measures 7 cm by 2 cm by 5 cm. Find its volume.
Answer: 70 cubic centimetres, since 7 x 2 x 5 = 70.
3. How many unit cubes exactly fill a box 3 units by 3 units by 3 units?
Answer: 27 unit cubes, since 3 x 3 x 3 = 27.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 packing unit cubes concretely (section 1), Lesson 2 counting by layers (section 2), Lesson 3 the formula (section 3), Lesson 4 mixed practice, real containers, and the exit ticket.
- Language to keep saying: volume is packed cubes not flat squares, count one layer then multiply by the number of layers, any order of multiplying the three edges gives the same volume. These pre-empt the three main misconceptions.
- This unit assumes fluent two- and three-factor multiplication. If 40 x 20 x 25 is a struggle, pause and revisit multiplication facts and multi-digit multiplication before pushing on to larger volumes.
- Curriculum note: the US formalises volume by unit cubes and the l x w x h formula at Grade 5. ACARA builds the prerequisite picture (parallel cross-sections stacking into a prism) at Year 5 (AC9M5SP01) but does not introduce the formal volume formula until Year 7 (AC9M7M02), so this unit runs roughly two years ahead of the Australian placement for the formula itself.
- Present mode and print both work: use Present to build a layered prism live with the class, then print for independent practice with the Print button.