Volume of 3D Shapes

Volume measures how much space a 3D object takes up. This lesson covers the volume formulas for cubes, rectangular prisms, cylinders, and triangular prisms.

What Is Volume?

Volume is the amount of three-dimensional space a solid occupies. It is measured in cubic units (cm³, m³).

Rectangular Prism (Cuboid)

Volume of a Cuboid

V = l \times w \times h

l = length, w = width, h = height.

A box 4 cm × 3 cm × 5 cm: $V = 4 \times 3 \times 5 = 60 \text{ cm}^3$

Cube

A special cuboid where $l = w = h = a$ :

Volume of a Cube

V = a^3

a = side length.

Cylinder

Volume of a Cylinder

V = \pi r^2 h

r = radius of circular base, h = height.

Triangular Prism

Volume of a Triangular Prism

V = \frac{1}{2} \times b \times h_{tri} \times l

b = base of triangle, h_tri = height of triangle, l = length of prism.

General Rule

For any prism: $V = A_{base} \times h$

where $A_{base}$ is the area of the cross-section (base) and $h$ is the length/height of the prism.

Tip Always check units! If dimensions are in cm, volume will be in cm³. Convert before calculating.

1
Identify the shape
Is it a cuboid, cylinder, prism, or something else?
2
Find the base area
Use the right formula for the cross-sectional shape.
3
Multiply by the height/length
V = A_base × h
4
Write the unit
Always include cm³, m³, etc.

Find the volume of a cylinder with radius 5 cm and height 12 cm.

$V = \pi \times 5^2 \times 12 = \pi \times 25 \times 12 = 300\pi \approx 942.5 \text{ cm}^3$

Real world: filling a swimming pool

A rectangular pool is 25 m long, 10 m wide, and 1.8 m deep. $V = 25 \times 10 \times 1.8 = 450\text{ m}^3$ . Since 1 m³ = 1000 litres, the pool holds 450 000 litres — vital information for the pump and chemical dosing systems.

Real world: packaging design

A cylindrical tin of beans has radius 4 cm and height 11 cm. $V = \pi \times 4^2 \times 11 \approx 553\text{ cm}^3$ . Food manufacturers compute this volume to know how much product fits in each tin and to minimise material costs.

Real world: cooking measurements

A triangular prism-shaped butter block is 10 cm long with a triangular cross-section of base 5 cm and height 4 cm. $V = \frac{1}{2} \times 5 \times 4 \times 10 = 100\text{ cm}^3 = 100\text{ mL}$ . Volume calculations link solid shapes directly to liquid measurements in cooking.

Tip For any prism, find the area of the cross-section first — the rest is just one multiplication:

V = A_{\text{base}} \times h

Practice Problems

Find the volume of a cuboid 7 cm × 4 cm × 3 cm.

Show answer

V = 7 × 4 × 3 = 84 cm³.

A cylinder has radius 3 cm and height 10 cm. Find its volume.

Show answer

V = π × 9 × 10 = 90π ≈ 282.7 cm³.

A cube has volume 125 cm³. What is its side length?

Show answer

a³ = 125 → a = 5 cm.