Cylinders

What you'll learn

How to use circle facts to find the volume of a cylinder.
How to find the total surface area of a closed cylinder.
How to handle semi-cylinders and simple combined solids.
How to solve reverse questions and practical unit-conversion problems.

1. The circle facts you need first

A cylinder is built from circles, so the key skills come from circle area and circumference.

Definition

Cylinder

A cylinder is a 3D solid with two equal, parallel circular faces joined by one curved surface. The radius is the distance from the centre of a circle to its edge, and the diameter is the distance all the way across the circle through the centre.

For a circle:

Diameter is twice the radius, so $d = 2 r$ .
Radius is half the diameter, so $\frac{d}{2}$ .
Area of a circle is $\pi r^2$ .
Circumference means the distance around the circle, and $2\pi r$ or $\pi d$ .

Common Mistake

Using the diameter as the radius

If a question gives the diameter, halve it before using formulas with $r$ . A diameter of 9 cm means the radius is 4.5 cm, not 9 cm.

Example

Finding the radius and circular area

A circular end of a cylinder has diameter 12 cm. Find its radius and the area of the circular end in terms of $π\pi$ .

Halve the diameter to get the radius.
$\frac{12}{2} = 6$
Use the circle area formula.
$\pi r^2 = \pi \times 6^2$
Square the radius.
$36\pi$
The area of the circular end is $36π36\pi$ cm².

2. Volume of a cylinder

Volume tells you how much space a 3D solid takes up. For a cylinder, imagine stacking identical circular layers on top of each other.

Key Idea

Cylinder volume

The volume of a cylinder is the area of the circular base multiplied by the height, so $\pi r^2h$ .

Here, $V$ means volume, $r$ means radius, and $h$ means height.

Example

Volume in terms of $π$

A cylinder has radius 4 cm and height 10 cm. Work out its volume in terms of $π\pi$ .

Write down the formula.
$\pi r^2h$
Substitute radius 4 and height 10.
$\pi \times 4^2 \times 10$
Square the radius, then multiply by the height.
$\pi \times 16 \times 10 = 160\pi$
The volume is $160π160\pi$ cm³.

Tip

When the answer says in terms of $π$

Leave $π\pi$ in your answer. Do not press the calculator’s $π\pi$ button unless the question asks for a decimal answer.

Example

Volume with a diameter and rounding

A cylinder has diameter 8 cm and height 11.5 cm. Work out its volume correct to 1 decimal place.

Find the radius by halving the diameter.
$\frac{8}{2} = 4$
Use the volume formula.
$\pi \times 4^2 \times 11.5$
Calculate the value.
$184\pi \approx 578.053$
Round to 1 decimal place: the volume is 578.1 cm³.

3. Total surface area of a cylinder

Surface area is the total area of the outside faces of a 3D shape.

A net is a flat layout of a 3D shape. A closed cylinder’s net has:

two circles, one for the top and one for the bottom;
one rectangle, which wraps around to make the curved surface.

The rectangle’s height is the cylinder height. Its width is the circumference of the circular base.

Diagram showing a cylinder and its net with two circular ends and one rectangle labelled using radius and height

For a closed cylinder:

area of two circles is $2πr22\pi r^2$ ;
curved surface area is $2πrh2\pi rh$ ;
total surface area is $2πr2+2πrh2\pi r^2 + 2\pi rh$ .

Common Mistake

Closed or open cylinder?

The formula $2πr2+2πrh2\pi r^2 + 2\pi rh$ is for a closed cylinder with both a top and a bottom. If a tank is open at the top, you would not include the top circle.

Example

Total surface area from radius and height

A closed cylinder has radius 3 cm and height 9.5 cm. Find its total surface area correct to 1 decimal place.

Start with the closed-cylinder surface area formula.
$TSA=2πr2+2πrh\text{TSA} = 2\pi r^2 + 2\pi rh$
Substitute radius 3 and height 9.5.
$TSA=2π×32+2π×3×9.5\text{TSA} = 2\pi \times 3^2 + 2\pi \times 3 \times 9.5$
Work out each part.
$TSA=18π+57π=75π\text{TSA} = 18\pi + 57\pi = 75\pi$
Convert to a decimal.
$75π≈235.61975\pi \approx 235.619$
The total surface area is 235.6 cm² to 1 decimal place.

Example

Total surface area from diameter

A solid cylinder has diameter 10 cm and height 13 cm. Work out its total surface area correct to 3 significant figures.

Halve the diameter to find the radius.
$\frac{10}{2} = 5$
Substitute into the total surface area formula.
$TSA=2π×52+2π×5×13\text{TSA} = 2\pi \times 5^2 + 2\pi \times 5 \times 13$
Simplify.
$TSA=50π+130π=180π\text{TSA} = 50\pi + 130\pi = 180\pi$
Convert to a decimal and round.
$180π≈565.487180\pi \approx 565.487$
The total surface area is 565 cm² to 3 significant figures.

4. Semi-cylinders and combined solids

A semi-cylinder is half a cylinder, usually made by cutting a cylinder lengthways.

For volume, a semi-cylinder is simply half the volume of the full cylinder:

\frac{1}{2}\pi r^2L

Here, $L$ is the length of the semi-cylinder.

For total surface area, think about each outside face:

half the curved surface of a full cylinder;
one flat rectangle made by the cut;
two semicircular ends, which together make one full circle.

Diagram of a semi-cylinder labelled with radius, length, curved surface, semicircular ends, and flat rectangular face

Example

Volume and surface area of a semi-cylinder

A semi-cylinder has radius 10 cm and length 40 cm. Find its volume and total surface area, each correct to 3 significant figures.

For volume, use half of the full-cylinder volume.
$\frac{1}{2}\pi \times 10^2 \times 40$
Simplify the volume.
$2000\pi \approx 6283.185$
The volume is $6.28 \times 10^3$ cm³ to 3 significant figures.
For surface area, add the curved half, the flat rectangle, and the two semicircular ends.
$TSA=πrL+2rL+πr2\text{TSA} = \pi rL + 2rL + \pi r^2$
Substitute radius 10 and length 40.
$TSA=π×10×40+2×10×40+π×102\text{TSA} = \pi \times 10 \times 40 + 2 \times 10 \times 40 + \pi \times 10^2$
Simplify and round.
$TSA=500π+800≈2370.796\text{TSA} = 500\pi + 800 \approx 2370.796$
The total surface area is $2.37 \times 10^3$ cm² to 3 significant figures.

A composite solid is a shape made from two or more simpler 3D solids. For volume, split it into shapes you know.

Example

Half cylinder on a rectangular prism

A solid is made from a rectangular prism with a half-cylinder on top. It has width 4 m, total height 5 m, and length 9 m. Find the volume correct to 3 significant figures.

The half-cylinder sits across the width, so its diameter is 4 m and its radius is 2 m.
The rectangular prism height is the total height minus the radius of the half-cylinder.
$5 - 2 = 3$
Find the prism volume.
$\times 3 \times 9 = 108$
Find the semi-cylinder volume.
$12π×22×9=18π\frac{1}{2}\pi \times 2^2 \times 9 = 18\pi$
Add the two volumes.
$18\pi \approx 164.549$
The volume is 165 m³ to 3 significant figures.

5. Working backwards

Sometimes you are given the volume or surface area and asked to find a missing radius or height. Use the formula first, then solve the equation.

Example

Finding a missing radius from volume

A cylinder has height 8 m and volume $288π288\pi$ m³. Find its radius.

Start with the volume formula.
$\pi r^2h$
Substitute the known values.
$288π=πr2×8288\pi = \pi r^2 \times 8$
Divide both sides by $π\pi$ , then divide by 8.
$288 = 8r^2$
Solve for $r^2$ .
$r^2 = 36$
Square root to find the positive radius.
$r = 6$
The radius is 6 m.

Example

Finding a missing height from surface area

A closed cylinder has radius 4 m and total surface area $112π112\pi$ m². Find its height.

Use the total surface area formula.
$TSA=2πr2+2πrh\text{TSA} = 2\pi r^2 + 2\pi rh$
Substitute radius 4 and surface area $112π112\pi$ .
$112π=2π×42+2π×4×h112\pi = 2\pi \times 4^2 + 2\pi \times 4 \times h$
Simplify.
$112π=32π+8πh112\pi = 32\pi + 8\pi h$
Subtract $32π32\pi$ from both sides.
$80π=8πh80\pi = 8\pi h$
Divide by $8π8\pi$ .
$h = 10$
The height is 10 m.

6. Practical problems: units, paint, and water

Cylinder questions often involve real-life units. Check whether all measurements are in the same unit before substituting into a formula.

Useful conversions:

1 m = 100 cm
1 m² = 10000 cm²
1 m³ = 1000000 cm³
1 litre = 1000 cm³

Common Mistake

Confusing square and cubic units

Surface area uses square units such as cm² or m². Volume uses cubic units such as cm³ or m³. Also, 1500 cm² is not 15 m²; since 1 m² is 10000 cm², 1500 cm² is 0.15 m².

Example

Paint needed for closed cylindrical tanks

Four closed cylindrical tanks are to be painted. Each tank has diameter 50 cm and height 1.2 m. One pot of paint covers 3 m². How many pots are needed?

Convert the diameter to metres.
$m50\text{ cm} = 0.5\text{ m}$
Find the radius in metres.
$\frac{0.5}{2} = 0.25$
Find the surface area of one closed tank.
$TSA=2π×0.252+2π×0.25×1.2\text{TSA} = 2\pi \times 0.25^2 + 2\pi \times 0.25 \times 1.2$
Simplify.
$TSA=0.725π≈2.278\text{TSA} = 0.725\pi \approx 2.278$
Find the total area for four tanks.
$\times 2.278 \approx 9.112$
Divide by the coverage per pot.
$9.112 \div 3 \approx 3.037$
You must buy 4 pots, because 3 pots would not cover enough area.

Tip

Rounding for practical answers

If you are buying pots of paint, packets, tiles, or containers, round up to the next whole number. You cannot buy 3.037 pots of paint.

Example

Filling a cylindrical tank

A cylindrical tank has radius 10 cm and height 60 cm. Water flows in at 1.5 litres per minute. Find the time to fill the tank to the nearest minute.

Find the tank volume in cm³.
$\pi \times 10^2 \times 60 = 6000\pi$
Convert cm³ to litres by dividing by 1000.
$6000π÷1000=6π6000\pi \div 1000 = 6\pi$
Divide by the flow rate.
$6π÷1.5≈12.5666\pi \div 1.5 \approx 12.566$
Round to the nearest minute: it takes 13 minutes.

Exam technique

In the exam

Write the correct formula first, then substitute the numbers carefully.
If you are given a diameter, halve it before using any formula with $r$ .
Check units: surface area is squared, volume is cubed, and practical answers may need rounding up.

Self review

Check yourself

Can you explain why the curved surface area of a cylinder is $2πrh2\pi rh$ ?
If a cylinder has diameter 14 cm, what radius should you use in the formula?
How would you convert 18000 cm³ into litres?

Cylinders

What you'll learn

How to use circle facts to find the volume of a cylinder.
How to find the total surface area of a closed cylinder.
How to handle semi-cylinders and simple combined solids.
How to solve reverse questions and practical unit-conversion problems.

1. The circle facts you need first

A cylinder is built from circles, so the key skills come from circle area and circumference.

Definition

Cylinder

For a circle:

Diameter is twice the radius, so $d = 2 r$ .
Radius is half the diameter, so $\frac{d}{2}$ .
Area of a circle is $\pi r^2$ .
Circumference means the distance around the circle, and $2\pi r$ or $\pi d$ .

Common Mistake

Using the diameter as the radius

If a question gives the diameter, halve it before using formulas with $r$ . A diameter of 9 cm means the radius is 4.5 cm, not 9 cm.

Example

Finding the radius and circular area

A circular end of a cylinder has diameter 12 cm. Find its radius and the area of the circular end in terms of $π\pi$ .

Halve the diameter to get the radius.
$\frac{12}{2} = 6$
Use the circle area formula.
$\pi r^2 = \pi \times 6^2$
Square the radius.
$36\pi$
The area of the circular end is $36π36\pi$ cm².

2. Volume of a cylinder

Volume tells you how much space a 3D solid takes up. For a cylinder, imagine stacking identical circular layers on top of each other.

Key Idea

Cylinder volume

The volume of a cylinder is the area of the circular base multiplied by the height, so $\pi r^2h$ .

Here, $V$ means volume, $r$ means radius, and $h$ means height.

Example

Volume in terms of $π$

A cylinder has radius 4 cm and height 10 cm. Work out its volume in terms of $π\pi$ .

Write down the formula.
$\pi r^2h$
Substitute radius 4 and height 10.
$\pi \times 4^2 \times 10$
Square the radius, then multiply by the height.
$\pi \times 16 \times 10 = 160\pi$
The volume is $160π160\pi$ cm³.

Tip

When the answer says in terms of $π$

Leave $π\pi$ in your answer. Do not press the calculator’s $π\pi$ button unless the question asks for a decimal answer.

Example

Volume with a diameter and rounding

A cylinder has diameter 8 cm and height 11.5 cm. Work out its volume correct to 1 decimal place.

Find the radius by halving the diameter.
$\frac{8}{2} = 4$
Use the volume formula.
$\pi \times 4^2 \times 11.5$
Calculate the value.
$184\pi \approx 578.053$
Round to 1 decimal place: the volume is 578.1 cm³.

3. Total surface area of a cylinder

Surface area is the total area of the outside faces of a 3D shape.

A net is a flat layout of a 3D shape. A closed cylinder’s net has:

two circles, one for the top and one for the bottom;
one rectangle, which wraps around to make the curved surface.

The rectangle’s height is the cylinder height. Its width is the circumference of the circular base.

Diagram showing a cylinder and its net with two circular ends and one rectangle labelled using radius and height

For a closed cylinder:

area of two circles is $2πr22\pi r^2$ ;
curved surface area is $2πrh2\pi rh$ ;
total surface area is $2πr2+2πrh2\pi r^2 + 2\pi rh$ .

Common Mistake

Closed or open cylinder?

The formula $2πr2+2πrh2\pi r^2 + 2\pi rh$ is for a closed cylinder with both a top and a bottom. If a tank is open at the top, you would not include the top circle.

Example

Total surface area from radius and height

A closed cylinder has radius 3 cm and height 9.5 cm. Find its total surface area correct to 1 decimal place.

Start with the closed-cylinder surface area formula.
$TSA=2πr2+2πrh\text{TSA} = 2\pi r^2 + 2\pi rh$
Substitute radius 3 and height 9.5.
$TSA=2π×32+2π×3×9.5\text{TSA} = 2\pi \times 3^2 + 2\pi \times 3 \times 9.5$
Work out each part.
$TSA=18π+57π=75π\text{TSA} = 18\pi + 57\pi = 75\pi$
Convert to a decimal.
$75π≈235.61975\pi \approx 235.619$
The total surface area is 235.6 cm² to 1 decimal place.

Example

Total surface area from diameter

A solid cylinder has diameter 10 cm and height 13 cm. Work out its total surface area correct to 3 significant figures.

Halve the diameter to find the radius.
$\frac{10}{2} = 5$
Substitute into the total surface area formula.
$TSA=2π×52+2π×5×13\text{TSA} = 2\pi \times 5^2 + 2\pi \times 5 \times 13$
Simplify.
$TSA=50π+130π=180π\text{TSA} = 50\pi + 130\pi = 180\pi$
Convert to a decimal and round.
$180π≈565.487180\pi \approx 565.487$
The total surface area is 565 cm² to 3 significant figures.

4. Semi-cylinders and combined solids

A semi-cylinder is half a cylinder, usually made by cutting a cylinder lengthways.

For volume, a semi-cylinder is simply half the volume of the full cylinder:

\frac{1}{2}\pi r^2L

Here, $L$ is the length of the semi-cylinder.

For total surface area, think about each outside face:

half the curved surface of a full cylinder;
one flat rectangle made by the cut;
two semicircular ends, which together make one full circle.

Diagram of a semi-cylinder labelled with radius, length, curved surface, semicircular ends, and flat rectangular face

Example

Volume and surface area of a semi-cylinder

A semi-cylinder has radius 10 cm and length 40 cm. Find its volume and total surface area, each correct to 3 significant figures.

For volume, use half of the full-cylinder volume.
$\frac{1}{2}\pi \times 10^2 \times 40$
Simplify the volume.
$2000\pi \approx 6283.185$
The volume is $6.28 \times 10^3$ cm³ to 3 significant figures.
For surface area, add the curved half, the flat rectangle, and the two semicircular ends.
$TSA=πrL+2rL+πr2\text{TSA} = \pi rL + 2rL + \pi r^2$
Substitute radius 10 and length 40.
$TSA=π×10×40+2×10×40+π×102\text{TSA} = \pi \times 10 \times 40 + 2 \times 10 \times 40 + \pi \times 10^2$
Simplify and round.
$TSA=500π+800≈2370.796\text{TSA} = 500\pi + 800 \approx 2370.796$
The total surface area is $2.37 \times 10^3$ cm² to 3 significant figures.

A composite solid is a shape made from two or more simpler 3D solids. For volume, split it into shapes you know.

Example

Half cylinder on a rectangular prism

A solid is made from a rectangular prism with a half-cylinder on top. It has width 4 m, total height 5 m, and length 9 m. Find the volume correct to 3 significant figures.

The half-cylinder sits across the width, so its diameter is 4 m and its radius is 2 m.
The rectangular prism height is the total height minus the radius of the half-cylinder.
$5 - 2 = 3$
Find the prism volume.
$\times 3 \times 9 = 108$
Find the semi-cylinder volume.
$12π×22×9=18π\frac{1}{2}\pi \times 2^2 \times 9 = 18\pi$
Add the two volumes.
$18\pi \approx 164.549$
The volume is 165 m³ to 3 significant figures.

5. Working backwards

Sometimes you are given the volume or surface area and asked to find a missing radius or height. Use the formula first, then solve the equation.

Example

Finding a missing radius from volume

A cylinder has height 8 m and volume $288π288\pi$ m³. Find its radius.

Start with the volume formula.
$\pi r^2h$
Substitute the known values.
$288π=πr2×8288\pi = \pi r^2 \times 8$
Divide both sides by $π\pi$ , then divide by 8.
$288 = 8r^2$
Solve for $r^2$ .
$r^2 = 36$
Square root to find the positive radius.
$r = 6$
The radius is 6 m.

Example

Finding a missing height from surface area

A closed cylinder has radius 4 m and total surface area $112π112\pi$ m². Find its height.

Use the total surface area formula.
$TSA=2πr2+2πrh\text{TSA} = 2\pi r^2 + 2\pi rh$
Substitute radius 4 and surface area $112π112\pi$ .
$112π=2π×42+2π×4×h112\pi = 2\pi \times 4^2 + 2\pi \times 4 \times h$
Simplify.
$112π=32π+8πh112\pi = 32\pi + 8\pi h$
Subtract $32π32\pi$ from both sides.
$80π=8πh80\pi = 8\pi h$
Divide by $8π8\pi$ .
$h = 10$
The height is 10 m.

6. Practical problems: units, paint, and water

Cylinder questions often involve real-life units. Check whether all measurements are in the same unit before substituting into a formula.

Useful conversions:

1 m = 100 cm
1 m² = 10000 cm²
1 m³ = 1000000 cm³
1 litre = 1000 cm³

Common Mistake

Confusing square and cubic units

Surface area uses square units such as cm² or m². Volume uses cubic units such as cm³ or m³. Also, 1500 cm² is not 15 m²; since 1 m² is 10000 cm², 1500 cm² is 0.15 m².

Example

Paint needed for closed cylindrical tanks

Four closed cylindrical tanks are to be painted. Each tank has diameter 50 cm and height 1.2 m. One pot of paint covers 3 m². How many pots are needed?

Convert the diameter to metres.
$m50\text{ cm} = 0.5\text{ m}$
Find the radius in metres.
$\frac{0.5}{2} = 0.25$
Find the surface area of one closed tank.
$TSA=2π×0.252+2π×0.25×1.2\text{TSA} = 2\pi \times 0.25^2 + 2\pi \times 0.25 \times 1.2$
Simplify.
$TSA=0.725π≈2.278\text{TSA} = 0.725\pi \approx 2.278$
Find the total area for four tanks.
$\times 2.278 \approx 9.112$
Divide by the coverage per pot.
$9.112 \div 3 \approx 3.037$
You must buy 4 pots, because 3 pots would not cover enough area.

Tip

Rounding for practical answers

If you are buying pots of paint, packets, tiles, or containers, round up to the next whole number. You cannot buy 3.037 pots of paint.

Example

Filling a cylindrical tank

A cylindrical tank has radius 10 cm and height 60 cm. Water flows in at 1.5 litres per minute. Find the time to fill the tank to the nearest minute.

Find the tank volume in cm³.
$\pi \times 10^2 \times 60 = 6000\pi$
Convert cm³ to litres by dividing by 1000.
$6000π÷1000=6π6000\pi \div 1000 = 6\pi$
Divide by the flow rate.
$6π÷1.5≈12.5666\pi \div 1.5 \approx 12.566$
Round to the nearest minute: it takes 13 minutes.

Exam technique

In the exam

Write the correct formula first, then substitute the numbers carefully.
If you are given a diameter, halve it before using any formula with $r$ .
Check units: surface area is squared, volume is cubed, and practical answers may need rounding up.

Self review

Check yourself

Can you explain why the curved surface area of a cylinder is $2πrh2\pi rh$ ?
If a cylinder has diameter 14 cm, what radius should you use in the formula?
How would you convert 18000 cm³ into litres?

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a cylinder

4. Semi-cylinders and combined solids

5. Working backwards

6. Practical problems: units, paint, and water

Recap questions

You've reached the end

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a cylinder

4. Semi-cylinders and combined solids

5. Working backwards

6. Practical problems: units, paint, and water

Recap questions

You've reached the end

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a cylinder

4. Semi-cylinders and combined solids

5. Working backwards

6. Practical problems: units, paint, and water

Recap questions

You've reached the end

Cylinders Revision Guide

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a cylinder

4. Semi-cylinders and combined solids

5. Working backwards

6. Practical problems: units, paint, and water

Recap questions

You've reached the end

Cylinders Revision Guide