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Lesson overview

Geometry — 3D shapes and volume

Volume measures how much space a 3D shape occupies. For a cylinder: $V = \pi r^2 h$ (area of circular base times height). For a cone: $V = \frac{1}{3}\pi r^2 h$ (one-third of the cylinder volume with the same base and height). For a sphere: $V = \frac{4}{3}\pi r^3$ . Always check units: if radius is in cm, volume is in ${cm}^{3}$ . If radius is in m, volume is in ${m}^{3}$ .

Volume of a cylinder

Find the volume of a cylinder with radius 3 cm and height 8 cm.

The volume is the area of the circular base (π ${r}^{2}$ ) times the height. Use a calculator for the final numerical answer.

Volume of a cone

Find the volume of a cone with radius 4 cm and height 9 cm.

A cone is one-third the volume of a cylinder with the same base and height. Don't forget the $\frac{1}{3}$ .

Volume of a sphere

Find the volume of a sphere with radius 5 cm.

The formula uses $r^3$ , not $r^2$ . Remember the coefficient: $\frac{4}{3}$ .

Worked example — a cylindrical can holds liquid

A cylindrical can has radius 4 cm and height 15 cm. How many millilitres does it hold? (1 ${cm}^{3}$ = 1 mL)

A cylindrical can.

AnswerThe can holds approximately 754 mL

Identify the shape and values: it's a cylinder with r = 4 cm and h = 15 cm.
Use the formula for cylinder volume: $V = \pi r^2 h$ .
Substitute: $V = \pi \times 4^2 \times 15 = \pi \times 16 \times 15 = 240\pi$ ${cm}^{3}.$
Calculate: $240\pi ≈ 753.98$ ${cm}^{3}.$
Convert units: since 1 ${cm}^{3}$ = 1 mL, the can holds approximately 754 mL.

Try it

Use the correct formula for each shape. Always include units in your final answer.

Question 1

What is the formula for the volume of a cylinder?

💡 Volume = area of base (π ${r}^{2}$ ) × height.

Question 2

A cylinder has radius 2 cm and height 10 cm. Find the volume in ${cm}^{3}$ to 1 d.p.

💡 $V = \pi \times 2^2 \times 10 = 40\pi ≈ 125.7$ ${cm}^{3}.$

Question 3

A cone and a cylinder have the same radius and height. The cone's volume is:

💡 Volume of cone = $\frac{1}{3}\pi r^2 h$ , which is exactly $\frac{1}{3}$ of the cylinder's volume.

Question 4

A sphere has radius 3 cm. Find the volume in ${cm}^{3}$ to 1 d.p.

💡 $V = \frac{4}{3}\pi \times 3^3 = \frac{4}{3}\pi \times 27 = 36\pi ≈ 113.1$ ${cm}^{3}.$

Question 5

A cone has radius 5 cm and height 12 cm. Find the volume in ${cm}^{3}$ to 1 d.p.

💡 $V = \frac{1}{3}\pi \times 5^2 \times 12 = \frac{1}{3}\pi \times 300 = 100\pi ≈ 314.2$ ${cm}^{3}.$

Common mistakes

Watch for these when working through the lesson.

Forgetting the $\frac{1}{3}$ in the cone formula: the formula is $\frac{1}{3}\pi r^2 h$ , not $\pi r^2 h$ .
Using diameter instead of radius: always check whether you've been given radius or diameter. If diameter, divide by 2 first.
Confusing units: if radius is in cm, the volume is in ${cm}^{3}$ . If radius is in metres, the volume is in ${m}^{3}$ .

Volume of cylinders, cones and spheres with formulae given

Lesson overview

Worked example — a cylindrical can holds liquid

Try it

Common mistakes

Related topics

Independent practice will plug in here