What Is Volume?

Volume is the amount of three-dimensional space inside an object.

For example, volume can describe:

• The space inside a box.
• The amount of room inside a storage container.
• The size of a block of wood.
• The capacity of a rectangular swimming pool.
• The amount of concrete needed to fill a rectangular form.

Area measures a two-dimensional surface.

Volume measures a three-dimensional space.

A three-dimensional object has three dimensions:

length × width × height

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Rectangular Prisms

A rectangular prism is a three-dimensional object with rectangular faces.

Common examples include:

• Boxes.
• Bricks.
• Books.
• Shipping containers.
• Rooms.
• Aquariums.
• Storage containers.

A rectangular prism has:

• 6 faces
• 12 edges
• 8 vertices

A cube is a special type of rectangular prism in which all edges have the same length.

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Understanding Cubic Units

Volume is measured using cubic units.

Common units include:

• mm³ — cubic millimetres
• cm³ — cubic centimetres
• m³ — cubic metres

A cubic centimetre, written:

1 cm³

is the volume of a cube measuring:

1 cm × 1 cm × 1 cm

Similarly:

1 m³

is the volume of a cube measuring:

1 m × 1 m × 1 m

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Why Do We Use Cubic Units?

Area has two dimensions:

length × width

so its units are squared:

cm × cm = cm²

Volume has three dimensions:

length × width × height

so its units are cubed:

cm × cm × cm = cm³

This gives us an important distinction:

Length → cm

Area → cm²

Volume → cm³

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Counting Cubes

Imagine a rectangular prism built from 1 cm³ cubes.

Suppose the bottom layer contains:

5 cubes × 4 cubes

That gives:

20 cubes per layer

If there are:

3 layers

then:

20 × 3 = 60 cubes

Therefore:

Volume = 60 cm³

This explains why we multiply three dimensions when calculating volume.

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Volume of a Rectangular Prism

The formula is:

V = l × w × h

where:

V = volume

l = length

w = width

h = height

Worked Example

A rectangular prism has:

length = 8 cm

width = 5 cm

height = 3 cm

Calculate:

V = 8 × 5 × 3

V = 120 cm³

The volume is:

120 cm³

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Thinking in Layers

We can also understand the calculation in two steps.

Suppose a prism measures:

8 cm × 5 cm × 3 cm

First find the area of the base:

8 × 5 = 40 cm²

Each layer therefore contains:

40 cubic centimetres per centimetre of height

There are three 1 cm-high layers:

40 × 3 = 120 cm³

So:

V = 120 cm³

This gives another useful way to think about volume:

Volume = area of base × height

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Worked Example: A Box

A box measures:

12 cm long

6 cm wide

5 cm high

Find its volume.

V = l × w × h

V = 12 × 6 × 5

V = 360 cm³

Therefore, the volume is:

360 cm³

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Worked Example: A Large Container

A rectangular container measures:

2 m × 1.5 m × 1 m

Calculate its volume.

V = 2 × 1.5 × 1

V = 3 m³

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Volume of a Cube

A cube has equal length, width, and height.

If each side has length s:

V = s × s × s

or:

V = s³

Example

A cube has sides of:

4 cm

Therefore:

V = 4³

V = 4 × 4 × 4

V = 64 cm³

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Cubing a Number

The notation:

5³

means:

5 × 5 × 5

Therefore:

5³ = 125

Similarly:

3³ = 27

10³ = 1000

For a cube:

V = s³

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Finding a Missing Dimension

Sometimes the volume is known, but one dimension is missing.

Because:

V = l × w × h

we can divide by the two known dimensions.

Worked Example

A rectangular prism has:

V = 240 cm³

l = 10 cm

w = 6 cm

Find the height.

240 = 10 × 6 × h

240 = 60h

Therefore:

h = 240 ÷ 60

h = 4 cm

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Another Missing-Dimension Example

A storage box has a volume of:

6000 cm³

Its width is:

20 cm

and its height is:

10 cm

Find its length.

V = l × w × h

6000 = l × 20 × 10

6000 = 200l

l = 6000 ÷ 200

l = 30 cm

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Finding the Side Length of a Cube

Suppose a cube has a volume of:

125 cm³

We need a number that multiplied by itself three times equals 125.

5 × 5 × 5 = 125

Therefore:

side length = 5 cm

The operation that reverses cubing is called the cube root.

∛125 = 5

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Comparing Volumes

We can compare objects by calculating their volumes.

Box A

Dimensions:

10 cm × 4 cm × 5 cm

Volume:

V = 10 × 4 × 5

V = 200 cm³

Box B

Dimensions:

8 cm × 6 cm × 5 cm

Volume:

V = 8 × 6 × 5

V = 240 cm³

Therefore:

Box B has the greater volume.

Difference:

240 − 200 = 40 cm³

Box B has 40 cm³ more volume than Box A.

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Comparing Volumes Using Ratios

Suppose:

Box A = 200 cm³

Box B = 600 cm³

Then:

600 ÷ 200 = 3

Box B has three times the volume of Box A.

This can be useful when comparing containers or designs.

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Units Must Match

Before multiplying dimensions, make sure all measurements use the same unit.

Suppose a box measures:

2 m × 50 cm × 40 cm

We should not immediately multiply:

2 × 50 × 40

because the units are different.

Convert to metres:

50 cm = 0.5 m

40 cm = 0.4 m

Now:

V = 2 × 0.5 × 0.4

V = 0.4 m³

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Converting Cubic Units

Volume conversions require special care.

We know:

1 m = 100 cm

Therefore:

1 m³ = 100 cm × 100 cm × 100 cm

So:

1 m³ = 1,000,000 cm³

This is very different from:

1 m = 100 cm

because volume has three dimensions.

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Understanding the Conversion

Imagine a cube measuring:

1 m × 1 m × 1 m

Convert each dimension to centimetres:

100 cm × 100 cm × 100 cm

Multiply:

100 × 100 × 100 = 1,000,000

Therefore:

1 m³ = 1,000,000 cm³

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Cubic Centimetres and Millilitres

An important relationship is:

1 cm³ = 1 mL

Therefore:

250 cm³ = 250 mL

and:

500 cm³ = 500 mL

This connects geometric volume with liquid volume.

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Cubic Decimetres and Litres

Another useful relationship is:

1 dm³ = 1 L

Because:

10 cm × 10 cm × 10 cm = 1000 cm³

we also have:

1 L = 1000 cm³

Therefore:

1000 cm³ = 1 L

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Worked Example: Container Capacity

A rectangular container measures:

20 cm × 10 cm × 15 cm

Volume:

V = 20 × 10 × 15

V = 3000 cm³

Since:

1000 cm³ = 1 L

then:

3000 cm³ = 3 L

The container has a theoretical internal capacity of:

3 L

assuming the given measurements are its internal dimensions.

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Internal and External Dimensions

In real-world container problems, it matters whether dimensions describe the inside or outside.

Suppose a wooden box has thick walls.

The outside might measure:

40 cm × 30 cm × 20 cm

but the inside dimensions could be smaller.

If you want to know how much the box can hold, you need the internal dimensions.

If you calculate using external dimensions, you may overestimate its capacity.

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Practical Example: Aquarium

An aquarium has internal dimensions:

80 cm × 30 cm × 40 cm

Volume:

V = 80 × 30 × 40

V = 96,000 cm³

Convert to litres:

96,000 ÷ 1000 = 96 L

The aquarium has a theoretical capacity of:

96 L

if filled completely.

In practice, an aquarium is usually not filled exactly to the top and may contain decorations, gravel, or equipment.

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Practical Example: Swimming Pool

A rectangular swimming pool measures:

10 m × 4 m × 1.5 m

Volume:

V = 10 × 4 × 1.5

V = 60 m³

Since:

1 m³ = 1000 L

the pool could contain:

60 × 1000 = 60,000 L

of water if filled to those dimensions.

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Practical Example: Storage Box

A storage box measures:

60 cm × 40 cm × 30 cm

Volume:

V = 60 × 40 × 30

V = 72,000 cm³

Convert to litres:

72,000 ÷ 1000 = 72 L

The box has a geometric volume of:

72 L

if those measurements represent its usable internal dimensions.

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Practical Example: Concrete

A rectangular concrete slab measures:

5 m long

3 m wide

0.2 m thick

Volume:

V = 5 × 3 × 0.2

V = 3 m³

Therefore:

3 m³ of concrete

is required in an ideal calculation.

In real construction, additional material may be ordered to account for waste and other factors.

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Practical Example: Soil

A rectangular garden bed measures:

4 m × 2 m

and needs soil to a depth of:

0.3 m

Volume:

V = 4 × 2 × 0.3

V = 2.4 m³

Therefore, approximately:

2.4 m³ of soil

is required before considering compaction or waste.

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Practical Example: Shipping

A shipping company needs to know how much space boxes occupy.

Suppose each box measures:

0.5 m × 0.4 m × 0.3 m

Volume of one box:

V = 0.5 × 0.4 × 0.3

V = 0.06 m³

For 20 boxes:

20 × 0.06 = 1.2 m³

The boxes occupy a combined geometric volume of:

1.2 m³

if considered individually.

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Volume and Surface Area Are Different

Volume measures the space inside an object.

Surface area measures the total area covering the outside surfaces.

Consider a cardboard box.

The amount the box can hold relates to its:

volume

The amount of cardboard needed to construct it relates to its:

surface area

These are different measurements.

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Length, Area, and Volume

It is important to distinguish between these three measurements.

Think:

Length → line

Area → surface

Volume → space

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How Changing Dimensions Changes Volume

Suppose a rectangular prism measures:

2 cm × 3 cm × 4 cm

Volume:

V = 24 cm³

Now double the length:

4 cm × 3 cm × 4 cm

Volume:

V = 48 cm³

Doubling one dimension doubles the volume.

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Doubling Two Dimensions

Start with:

2 cm × 3 cm × 4 cm

Volume:

24 cm³

Double the length and width:

4 cm × 6 cm × 4 cm

Volume:

96 cm³

The volume becomes:

4 times larger

because:

2 × 2 = 4

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Doubling Every Dimension

Consider a cube with side:

2 cm

Volume:

2³ = 8 cm³

Double the side length to:

4 cm

New volume:

4³ = 64 cm³

Compare:

64 ÷ 8 = 8

Therefore, doubling all three dimensions makes the volume:

8 times larger

because:

2 × 2 × 2 = 8

This is an important property of three-dimensional objects.

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Tripling Every Dimension

Suppose every dimension is multiplied by:

3

Then the volume is multiplied by:

3³

= 27

So an object enlarged to three times its original length, width, and height has:

27 times the original volume

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Estimating Volume

Estimation can help determine whether an answer is reasonable.

Suppose a box measures:

9.8 cm × 5.1 cm × 4.2 cm

Estimate:

10 × 5 × 4

≈ 200 cm³

The exact calculation is approximately:

209.9 cm³

The exact answer is close to the estimate, so it appears reasonable.

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Composite Rectangular Prisms

Some three-dimensional objects can be divided into several rectangular prisms.

To calculate the total volume:

1. Divide the object into rectangular prisms.
2. Calculate each volume.
3. Add the volumes.

For example:

Prism A = 120 cm³

Prism B = 80 cm³

Then:

Total volume = 120 + 80

Total volume = 200 cm³

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Subtracting Volumes

Sometimes a section is removed from a larger rectangular prism.

Suppose:

large prism = 500 cm³

and:

removed section = 120 cm³

Then:

remaining volume = 500 − 120

remaining volume = 380 cm³

This is similar to subtracting areas when working with composite two-dimensional shapes.

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A Reliable Problem-Solving Method

When solving a volume problem:

Step 1: Identify the three dimensions.

Step 2: Check that all dimensions use compatible units.

Step 3: Use:

V = l × w × h

Step 4: Multiply carefully.

Step 5: Write the answer using cubic units.

Step 6: Convert the volume if necessary.

Step 7: Check whether the answer is reasonable.

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Common Mistakes

Using Square Units

Incorrect:

V = 120 cm²

Correct:

V = 120 cm³

Volume requires cubic units.

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Forgetting the Third Dimension

Incorrect:

V = length × width

That calculates the area of a rectangular face.

Correct:

V = length × width × height

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Adding the Dimensions

For a box measuring:

5 cm × 4 cm × 3 cm

Incorrect:

5 + 4 + 3 = 12 cm³

Correct:

5 × 4 × 3 = 60 cm³

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Mixing Units

Do not multiply:

2 m × 50 cm × 30 cm

without converting the dimensions to compatible units first.

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Confusing cm³ and mL

Remember:

1 cm³ = 1 mL

This is a direct equivalence.

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Confusing m³ and Litres

Remember:

1 m³ = 1000 L

not:

1 m³ = 1 L

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Converting Cubic Units Like Length Units

Remember:

1 m = 100 cm

but:

1 m³ = 1,000,000 cm³

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Using External Dimensions for Capacity

If a container has thick walls, its internal volume is smaller than the volume calculated from its external dimensions.

Use internal measurements when determining capacity.

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Confusing Volume and Surface Area

Volume = space inside

Surface area = area covering the outside

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Check Your Understanding

1. Define volume.

2. Why is volume measured in cubic units?

3. What does 1 cm³ represent?

4. Write the formula for the volume of a rectangular prism.

5. Find the volume of a prism measuring 8 cm × 5 cm × 3 cm.

6. Find the volume of a box measuring 12 cm × 10 cm × 6 cm.

7. Find the volume of a rectangular prism measuring 4 m × 3 m × 2 m.

8. Find the volume of a cube with side length 5 cm.

9. A prism has a volume of 240 cm³, a length of 10 cm, and a width of 6 cm. Find its height.

10. A box has a volume of 900 cm³. Its width is 10 cm and its height is 9 cm. Find its length.

11. A cube has a volume of 216 cm³. Find its side length.

12. Box A measures 10 cm × 6 cm × 4 cm. Box B measures 8 cm × 8 cm × 5 cm. Which has the greater volume?

13. By how much do the volumes in Question 12 differ?

14. Convert 4000 cm³ to litres.

15. Convert 7.5 L to cm³.

16. Explain why 1 m³ equals 1,000,000 cm³.

17. An aquarium measures internally 60 cm × 30 cm × 40 cm. Find its volume in cm³.

18. Convert your answer to Question 17 into litres.

19. A rectangular swimming pool measures 8 m × 5 m × 2 m. Find its volume.

20. Convert the pool volume in Question 19 to litres.

21. A garden bed measures 5 m × 2 m and needs soil 0.4 m deep. How much soil is required?

22. A concrete slab measures 6 m × 4 m × 0.15 m. Find the volume of concrete required.

23. A rectangular prism has a volume of 360 cm³. Its length is 12 cm and width is 5 cm. Find its height.

24. Explain the difference between area and volume.

25. Explain the difference between volume and surface area.

26. A box's length is doubled while its width and height remain unchanged. What happens to its volume?

27. Every dimension of a rectangular prism is doubled. By what factor does its volume increase?

28. Every dimension of a cube is tripled. By what factor does its volume increase?

29. A storage container has external dimensions of 50 cm × 40 cm × 30 cm. Explain why these dimensions may not give its actual storage capacity.

30. Give three real-world situations in which calculating volume would be useful.

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Key Terms

• Volume – amount of three-dimensional space inside an object.
• Rectangular prism – three-dimensional object with six rectangular faces.
• Cube – rectangular prism in which all edges have equal length.
• Cubic unit – unit used to measure volume.
• Cubic centimetre (cm³) – volume of a cube measuring 1 cm × 1 cm × 1 cm.
• Cubic metre (m³) – volume of a cube measuring 1 m × 1 m × 1 m.
• Length – one dimension of an object.
• Width – second dimension perpendicular to length.
• Height – third dimension perpendicular to length and width.
• Capacity – amount that a container can hold.
• Composite solid – three-dimensional object made from two or more simpler solids.
• Surface area – total area covering the outside surfaces of a three-dimensional object.

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Key Takeaways

• Volume measures the amount of three-dimensional space inside an object.
• Volume is measured using cubic units such as cm³ and m³.
• A cubic centimetre is a cube measuring 1 cm × 1 cm × 1 cm.
• The volume of a rectangular prism is V = l × w × h.
• The volume of a cube is V = s³.
• Volume can also be understood as area of base × height.
• All dimensions must use compatible units before calculating volume.
• 1 cm³ = 1 mL.
• 1000 cm³ = 1 L.
• 1 m³ = 1000 L.
• 1 m³ = 1,000,000 cm³.
• Missing dimensions can be found by dividing the volume by the known dimensions.
• Volumes can be compared using differences or ratios.
• Doubling one dimension doubles the volume if the other dimensions remain unchanged.
• Doubling all three dimensions increases volume by a factor of 8.
• Tripling all three dimensions increases volume by a factor of 27.
• Internal dimensions should be used when calculating the capacity of a container.
• Volume calculations are useful for storage, aquariums, pools, concrete, soil, packaging, and construction.

The central idea is:

Volume = length × width × height

or, more generally:

Volume = area of base × height.