What Is Volume?
In maths, volume is the amount of three-dimensional space an object occupies. A rectangular box measuring 10 cm × 4 cm × 5 cm has a volume of 200 cubic centimetres (10 × 4 × 5 = 200 cm³).
Volume applies only to 3D objects. A flat shape — a square, a circle, a triangle — has area, not volume.
The Volume Formula
For most regular 3D shapes, volume equals the area of the base multiplied by the height.
V = A × h
where A is the area of the base and h is the height of the shape.
This rule covers cubes, cuboids, cylinders, and prisms. Three shapes follow related but distinct formulas:
A cone has one-third the volume of a cylinder with the same base and height.
A pyramid has one-third the volume of a prism with the same base and height.
A sphere uses V = (4/3)πr³, derived through integration.
Why Volume Is Measured in Cubic Units
Volume counts how many unit cubes fit inside an object. A unit cube has all sides of length 1 — for example, 1 cm. A cuboid that fits 27 such cubes has a volume of 27 cm³.
The cubic unit notation (cm³, m³, mm³) follows from this idea. A cube of side 1 cm has length 1 cm, width 1 cm, and height 1 cm, giving 1 × 1 × 1 = 1 cm³. The exponent 3 records that three dimensions have been multiplied together.
Units of Volume
SI and Metric Units
Unit | Symbol | Conversion |
|---|---|---|
Cubic millimetre | mm³ | 1,000 mm³ = 1 cm³ |
Cubic centimetre | cm³ | 1,000,000 cm³ = 1 m³ |
Cubic metre | m³ | 1 m³ = 1,000 litres |
Cubic kilometre | km³ | 1 km³ = 1,000,000,000 m³ |
The cubic metre (m³) is the SI derived unit of volume.
Capacity Units (Litres and Gallons)
Capacity is the amount of liquid or gas a container can hold. Capacity units convert directly to cubic units.
Capacity | Cubic Equivalent |
|---|---|
1 millilitre (mL) | 1 cm³ |
1 litre (L) | 1,000 cm³ |
1 litre (L) | 0.001 m³ |
1 US gallon | 3.785 litres |
1 UK (imperial) gallon | 4.546 litres |
The relationship 1 litre = 1,000 cm³ comes from geometry. A cube with sides of 10 cm has a volume of 10 × 10 × 10 = 1,000 cm³, and that volume is defined as 1 litre.
Volume Formulas for Common 3D Shapes
The table below lists the standard volume formulas for common 3D shapes.
Shape | Formula | Variables |
|---|---|---|
Cube | V = a³ | a = side length |
Cuboid (rectangular prism) | V = l × b × h | l, b, h = length, breadth, height |
Cylinder | V = πr²h | r = base radius, h = height |
Cone | V = (1/3)πr²h | r = base radius, h = height |
Sphere | V = (4/3)πr³ | r = radius |
Hemisphere | V = (2/3)πr³ | r = radius |
Prism (general) | V = base area × height | depends on base shape |
Pyramid | V = (1/3) × base area × height | depends on base shape |
Cube
V = a³. All three dimensions of a cube are equal, so the formula simplifies from l × b × h to a × a × a. A die with a side of 2 cm has a volume of 8 cm³.
Cuboid (Rectangular Prism)
V = l × b × h. The base is a rectangle of area l × b, and the height is h. Cuboids include cardboard boxes, books, bricks, and most rectangular containers.
Cylinder
V = πr²h. The base is a circle with area πr², and h is the height. The same logic — base area times height — applies as it does to a prism.
Cone
V = (1/3)πr²h. A cone's volume is exactly one-third of a cylinder with the same base radius and height. This relationship can be checked by filling a cone-shaped container with water and pouring it into a matching cylinder — three full cones fill the cylinder.
Sphere
V = (4/3)πr³, where r is the radius. The formula is derived through integration in higher mathematics and is taken as standard at school level.
Hemisphere
V = (2/3)πr³. A hemisphere is half of a sphere, so its volume is half of (4/3)πr³.
Prism
The volume of any prism is base area × height. The base can be a triangle, hexagon, or any polygon, and the area calculation adapts to the shape of that base.
Pyramid
V = (1/3) × base area × height. A pyramid's volume is one-third of a prism with the same base and height — the same factor of one-third that applies to a cone relative to a cylinder.
Worked Examples
Example 1 — Volume of a Cuboid
A rectangular box has length 8 cm, breadth 5 cm, and height 3 cm. Find its volume.
V = l × b × h V = 8 × 5 × 3 V = 120
Volume = 120 cm³
Example 2 — Volume of a Cylinder
A cylindrical can has a base radius of 7 cm and a height of 10 cm. Find its volume. Use π = 22/7.
V = πr²h V = (22/7) × 7² × 10 V = (22/7) × 49 × 10 V = 22 × 7 × 10 V = 1,540
Volume = 1,540 cm³
Example 3 — Volume of a Sphere
A spherical ball has a radius of 6 cm. Find its volume. Use π = 3.14.
V = (4/3)πr³ V = (4/3) × 3.14 × 6³ V = (4/3) × 3.14 × 216 V = (4 × 3.14 × 216) / 3 V = 2,712.96 / 3 V = 904.32
Volume = 904.32 cm³
Volume vs Area vs Capacity
The three terms are related but distinct.
Property | What It Measures | Applies To | Units | Example |
|---|---|---|---|---|
Area | 2D space covered | Flat shapes (rectangle, circle, triangle) | Square units (cm², m²) | The floor area of a room |
Volume | 3D space occupied | Solid objects | Cubic units (cm³, m³) | The space a brick takes up |
Capacity | Amount a container can hold | Hollow objects | Litres, millilitres, gallons | The water a bottle holds |
Volume and capacity are often used interchangeably, but the precise distinction is this: volume describes the space an object itself occupies, and capacity describes how much a hollow container can hold inside it.
Volume of Irregular Shapes
Irregular shapes — objects without a defined formula — can be measured using the water displacement method. The object is submerged in a measured volume of water inside a graduated container. The rise in the water level equals the volume of the object.
Did you know? Around 250 BCE, Archimedes used the displacement method to determine whether a king's crown was made of pure gold. By submerging the crown in water and comparing the displaced volume to that of an equal weight of pure gold, he could check the crown's density without melting it down.
Common Mistakes
Three errors appear most often when students calculate volume.
Using square units instead of cubic units. Volume requires cubic units (cm³, m³). Writing the answer as cm² changes the meaning of the result and is marked wrong.
Mixing measurement units in one calculation. All dimensions must be in the same unit before multiplication. If length is in metres and breadth is in centimetres, one must be converted before applying the formula.
Using diameter instead of radius. Cylinder, cone, and sphere formulas all use the radius, not the diameter. If only the diameter is given, halve it before substituting into the formula.
Related Terms
Term | Meaning | How It Relates to Volume |
|---|---|---|
Area | The space covered by a 2D shape | Volume is the 3D extension of area |
Surface area | The total area of an object's outer faces | A different measurement of the same 3D object |
Capacity | The amount a container can hold | Often expressed as volume in litres or gallons |
Density | Mass per unit volume | Density = mass ÷ volume |
Cubic unit | A unit of length cubed (cm³, m³) | The standard unit of volume |
Litre | A unit of capacity equal to 1,000 cm³ | The volume of a cube with sides of 10 cm |
Displacement | The volume of fluid pushed aside by a submerged object | Used to find the volume of irregular shapes |
Where Volume Appears in the Curriculum
Volume is introduced in late primary and built on through middle school across major curricula.
NCERT (India): Class 9, Chapter 13 — Surface Areas and Volumes; revisited in Class 10.
CCSS (US): 5.MD.C.5 (Grade 5), 6.G.A.2 (Grade 6), 7.G.B.6 (Grade 7).
UK National Curriculum: Key Stage 2 (Year 6) and Key Stage 3.
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