Cylinders

What you'll learn

How radius, diameter and height describe a cylinder.
How to calculate the volume of a cylinder.
How to calculate the total surface area of a solid cylinder.
How to handle half-cylinders, missing lengths, paint and filling questions.

1. The circle facts you need first

Definition

Cylinder and circle words

A cylinder is a 3D solid with two equal, parallel circular faces joined by one curved surface.

A cylinder has two matching circular faces, a radius across each circular face, and a perpendicular height between the faces.

The radius is the distance from the centre of a circle to its edge.
The diameter is the distance all the way across a circle through the centre, so $d = 2 r$ .
The height is the perpendicular distance between the two circular faces.

Two circle formulas appear again and again:

Area of a circle: $A=πr2A=\pi r^2$
Circumference, which means distance around the circle: $C=2πrC=2\pi r$

Example

Choosing the radius

A cylinder has diameter 14 cm.

The radius is half of the diameter across the circular face.

The diameter goes all the way across the circle.
The radius is half the diameter:
$r=14÷2=7r=14\div 2=7$
So in any cylinder formula, use 7 cm for the radius.

Common Mistake

Diameter is not radius

If the question gives the diameter, halve it before using the formula. Squaring the diameter instead of the radius makes the answer much too large.

2. Volume of a cylinder

Volume means the amount of space inside a 3D shape. It is measured in cubic units such as cm³ or m³.

Key Idea

Volume formula

The volume of a cylinder is the area of the circular base multiplied by the height: $V=πr2hV=\pi r^2h$ .

Cylinder volume can be visualised as a stack of identical circular layers with base area πr² and height h.

Think of it as stacking lots of identical circles on top of each other.

Example

Volume from a diameter

A cylinder has diameter 8 cm and height 13 cm. Find its volume correct to 1 decimal place.

For volume, halve the diameter to get radius 4 cm, then use the perpendicular height of 13 cm.

Find the radius by halving the diameter:
$r=8÷2=4r=8\div 2=4$
Substitute into $V=πr2hV=\pi r^2h$ :
$V=π×42×13=208πV=\pi \times 4^2 \times 13=208\pi$
Use your calculator for the decimal value:
$208π=653.451…208\pi=653.451\ldots$
Rounded to 1 decimal place, the volume is 653.5 cm³.

Tip

In terms of pi

If the question says “give your answer in terms of $π\pi$ ”, leave $π\pi$ in your final answer. For the example above, you would write $208π208\pi$ cm³.

3. Total surface area of a solid cylinder

Surface area means the total area on the outside of a 3D shape.

For a closed solid cylinder, there are three outside parts:

the top circle
the bottom circle
the curved surface around the side

The curved surface unwraps into a rectangle. Its width is the circumference of the circle, and its height is the cylinder height.

The curved surface of a closed cylinder unwraps to a rectangle of width 2πr and height h, alongside the two circular ends.

So the total surface area formula is:

TSA=2πr2+2πrh\text{TSA}=2\pi r^2+2\pi rh

Example

Surface area from a diameter

A solid cylinder has diameter 10 cm and height 14 cm. Find the total surface area correct to 3 significant figures.

The total surface area includes both circular ends and the curved side of the closed cylinder.

Halve the diameter to get the radius:
$r=10÷2=5r=10\div 2=5$
Work out the two circular ends:
$2πr2=2π×52=50π2\pi r^2=2\pi \times 5^2=50\pi$
Work out the curved surface:
$2πrh=2π×5×14=140π2\pi rh=2\pi \times 5 \times 14=140\pi$
Add the parts together:
$50π+140π=190π50\pi+140\pi=190\pi$
Convert to a decimal and round:
$190π=596.902…190\pi=596.902\ldots$
The total surface area is 597 cm² to 3 significant figures.

Common Mistake

Forgetting the top and bottom

For total surface area, include both circular ends unless the question clearly says the cylinder is open.

4. Half-cylinders

A semi-cylinder is half a cylinder, cut lengthways.

For volume, halve the full cylinder volume. If the length is called $L$ , then:

V=12πr2LV=\frac{1}{2}\pi r^2L

For total surface area of a semi-cylinder, remember these parts:

half the curved surface
two semi-circular ends
the flat rectangular face made by the cut

Example

Volume and surface area of a semi-cylinder

A semi-cylinder has radius 6 cm and length 25 cm. Find the volume and total surface area correct to 3 significant figures.

A semi-cylinder is half a cylinder cut lengthways, with a curved half-surface, two semi-circular ends and a flat rectangular cut face.

Work out the volume using half the cylinder formula:
$V=12π×62×25=450πV=\frac{1}{2}\pi \times 6^2 \times 25=450\pi$
Convert and round: $450π=1413.7…450\pi=1413.7\ldots$ , so the volume is 1410 cm³ to 3 significant figures.
The half-curved surface area is:
$π×6×25=150π\pi \times 6 \times 25=150\pi$
The two semi-circular ends make one full circle:
$π×62=36π\pi \times 6^2=36\pi$
The flat rectangle is 12 cm by 25 cm, so its area is 300 cm².
Add the surface areas:
$150π+36π+300=186π+300=884.3…150\pi+36\pi+300=186\pi+300=884.3\ldots$
The total surface area is 884 cm² to 3 significant figures.

Tip

Combined solids

For a shape made from a cuboid and a half-cylinder, find each volume separately and add them. For surface area, do not count hidden faces where the shapes touch.

5. Working backwards

Sometimes the volume or surface area is given, and you must find a missing radius or height.

Use the same formula, substitute what you know, then solve.

Example

Finding a missing radius

A cylinder has height 10 m and volume $490π490\pi$ m³. Find the radius.

Start with the volume formula:
$V=πr2hV=\pi r^2h$
Substitute the given volume and height:
$490π=πr2×10490\pi=\pi r^2 \times 10$
Divide by $10π10\pi$ :
$r^2=49$
Square root both sides:
$r = 7$
The radius is 7 m.

Common Mistake

Stopping at r squared

If you find $r^2$ , you have not finished. Take the square root to find $r$ .

6. Units, paint and filling questions

Real-life cylinder questions often involve tanks, paint or water.

Painting questions usually need surface area.
Filling questions usually need volume.
Make sure all measurements are in the same units before calculating.
1 litre = 1000 cm³.
100 cm = 1 m, but 10,000 cm² = 1 m².

Example

Paint needed for cylindrical tanks

Four closed cylindrical tanks are painted. Each tank has diameter 50 cm and height 1.5 m. One pot of paint covers 3 m². Find the number of pots needed.

Painting four closed tanks means finding the outside surface area of each full cylinder, including top, bottom and curved side.

Convert the diameter to metres: 50 cm = 0.5 m.
Find the radius:
$r=0.5÷2=0.25r=0.5\div 2=0.25$
Find the surface area of one closed tank:
$TSA=2π(0.25)2+2π(0.25)(1.5)=0.875π\text{TSA}=2\pi(0.25)^2+2\pi(0.25)(1.5)=0.875\pi$
Convert to a decimal:
$0.875π=2.748…0.875\pi=2.748\ldots$
For four tanks:
$4×2.748…=10.995…4\times 2.748\ldots=10.995\ldots$
Divide by the coverage of one pot:
$10.995…÷3=3.665…10.995\ldots \div 3=3.665\ldots$
You must buy 4 pots, because 3 pots would not be enough.

Common Mistake

Area unit conversions

Do not say 1500 cm² is 15 m². Since 1 m² is 10,000 cm², 1500 cm² is only 0.15 m².

Exam technique

In the exam

Check whether you have been given radius or diameter before using any formula.
Write the formula first, then substitute the numbers carefully.
Keep calculator values unrounded until the final line, then round exactly as requested.

Self review

Check yourself

If a cylinder has diameter 18 cm, what radius should you use in the formula?
Which formula would you use for volume: $V=πr2hV=\pi r^2h$ or $TSA=2πr2+2πrh\text{TSA}=2\pi r^2+2\pi rh$ ?
In a painting question, why might you need to round the number of pots up?

Cylinders

What you'll learn

How radius, diameter and height describe a cylinder.
How to calculate the volume of a cylinder.
How to calculate the total surface area of a solid cylinder.
How to handle half-cylinders, missing lengths, paint and filling questions.

1. The circle facts you need first

Definition

Cylinder and circle words

A cylinder is a 3D solid with two equal, parallel circular faces joined by one curved surface.

A cylinder has two matching circular faces, a radius across each circular face, and a perpendicular height between the faces.

The radius is the distance from the centre of a circle to its edge.
The diameter is the distance all the way across a circle through the centre, so $d = 2 r$ .
The height is the perpendicular distance between the two circular faces.

Two circle formulas appear again and again:

Area of a circle: $A=πr2A=\pi r^2$
Circumference, which means distance around the circle: $C=2πrC=2\pi r$

Example

Choosing the radius

A cylinder has diameter 14 cm.

The radius is half of the diameter across the circular face.

The diameter goes all the way across the circle.
The radius is half the diameter:
$r=14÷2=7r=14\div 2=7$
So in any cylinder formula, use 7 cm for the radius.

Common Mistake

Diameter is not radius

If the question gives the diameter, halve it before using the formula. Squaring the diameter instead of the radius makes the answer much too large.

2. Volume of a cylinder

Volume means the amount of space inside a 3D shape. It is measured in cubic units such as cm³ or m³.

Key Idea

Volume formula

The volume of a cylinder is the area of the circular base multiplied by the height: $V=πr2hV=\pi r^2h$ .

Cylinder volume can be visualised as a stack of identical circular layers with base area πr² and height h.

Think of it as stacking lots of identical circles on top of each other.

Example

Volume from a diameter

A cylinder has diameter 8 cm and height 13 cm. Find its volume correct to 1 decimal place.

For volume, halve the diameter to get radius 4 cm, then use the perpendicular height of 13 cm.

Find the radius by halving the diameter:
$r=8÷2=4r=8\div 2=4$
Substitute into $V=πr2hV=\pi r^2h$ :
$V=π×42×13=208πV=\pi \times 4^2 \times 13=208\pi$
Use your calculator for the decimal value:
$208π=653.451…208\pi=653.451\ldots$
Rounded to 1 decimal place, the volume is 653.5 cm³.

Tip

In terms of pi

If the question says “give your answer in terms of $π\pi$ ”, leave $π\pi$ in your final answer. For the example above, you would write $208π208\pi$ cm³.

3. Total surface area of a solid cylinder

Surface area means the total area on the outside of a 3D shape.

For a closed solid cylinder, there are three outside parts:

the top circle
the bottom circle
the curved surface around the side

The curved surface unwraps into a rectangle. Its width is the circumference of the circle, and its height is the cylinder height.

The curved surface of a closed cylinder unwraps to a rectangle of width 2πr and height h, alongside the two circular ends.

So the total surface area formula is:

TSA=2πr2+2πrh\text{TSA}=2\pi r^2+2\pi rh

Example

Surface area from a diameter

A solid cylinder has diameter 10 cm and height 14 cm. Find the total surface area correct to 3 significant figures.

The total surface area includes both circular ends and the curved side of the closed cylinder.

Halve the diameter to get the radius:
$r=10÷2=5r=10\div 2=5$
Work out the two circular ends:
$2πr2=2π×52=50π2\pi r^2=2\pi \times 5^2=50\pi$
Work out the curved surface:
$2πrh=2π×5×14=140π2\pi rh=2\pi \times 5 \times 14=140\pi$
Add the parts together:
$50π+140π=190π50\pi+140\pi=190\pi$
Convert to a decimal and round:
$190π=596.902…190\pi=596.902\ldots$
The total surface area is 597 cm² to 3 significant figures.

Common Mistake

Forgetting the top and bottom

For total surface area, include both circular ends unless the question clearly says the cylinder is open.

4. Half-cylinders

A semi-cylinder is half a cylinder, cut lengthways.

For volume, halve the full cylinder volume. If the length is called $L$ , then:

V=12πr2LV=\frac{1}{2}\pi r^2L

For total surface area of a semi-cylinder, remember these parts:

half the curved surface
two semi-circular ends
the flat rectangular face made by the cut

Example

Volume and surface area of a semi-cylinder

A semi-cylinder has radius 6 cm and length 25 cm. Find the volume and total surface area correct to 3 significant figures.

A semi-cylinder is half a cylinder cut lengthways, with a curved half-surface, two semi-circular ends and a flat rectangular cut face.

Work out the volume using half the cylinder formula:
$V=12π×62×25=450πV=\frac{1}{2}\pi \times 6^2 \times 25=450\pi$
Convert and round: $450π=1413.7…450\pi=1413.7\ldots$ , so the volume is 1410 cm³ to 3 significant figures.
The half-curved surface area is:
$π×6×25=150π\pi \times 6 \times 25=150\pi$
The two semi-circular ends make one full circle:
$π×62=36π\pi \times 6^2=36\pi$
The flat rectangle is 12 cm by 25 cm, so its area is 300 cm².
Add the surface areas:
$150π+36π+300=186π+300=884.3…150\pi+36\pi+300=186\pi+300=884.3\ldots$
The total surface area is 884 cm² to 3 significant figures.

Tip

Combined solids

For a shape made from a cuboid and a half-cylinder, find each volume separately and add them. For surface area, do not count hidden faces where the shapes touch.

5. Working backwards

Sometimes the volume or surface area is given, and you must find a missing radius or height.

Use the same formula, substitute what you know, then solve.

Example

Finding a missing radius

A cylinder has height 10 m and volume $490π490\pi$ m³. Find the radius.

Start with the volume formula:
$V=πr2hV=\pi r^2h$
Substitute the given volume and height:
$490π=πr2×10490\pi=\pi r^2 \times 10$
Divide by $10π10\pi$ :
$r^2=49$
Square root both sides:
$r = 7$
The radius is 7 m.

Common Mistake

Stopping at r squared

If you find $r^2$ , you have not finished. Take the square root to find $r$ .

6. Units, paint and filling questions

Real-life cylinder questions often involve tanks, paint or water.

Painting questions usually need surface area.
Filling questions usually need volume.
Make sure all measurements are in the same units before calculating.
1 litre = 1000 cm³.
100 cm = 1 m, but 10,000 cm² = 1 m².

Example

Paint needed for cylindrical tanks

Four closed cylindrical tanks are painted. Each tank has diameter 50 cm and height 1.5 m. One pot of paint covers 3 m². Find the number of pots needed.

Painting four closed tanks means finding the outside surface area of each full cylinder, including top, bottom and curved side.

Convert the diameter to metres: 50 cm = 0.5 m.
Find the radius:
$r=0.5÷2=0.25r=0.5\div 2=0.25$
Find the surface area of one closed tank:
$TSA=2π(0.25)2+2π(0.25)(1.5)=0.875π\text{TSA}=2\pi(0.25)^2+2\pi(0.25)(1.5)=0.875\pi$
Convert to a decimal:
$0.875π=2.748…0.875\pi=2.748\ldots$
For four tanks:
$4×2.748…=10.995…4\times 2.748\ldots=10.995\ldots$
Divide by the coverage of one pot:
$10.995…÷3=3.665…10.995\ldots \div 3=3.665\ldots$
You must buy 4 pots, because 3 pots would not be enough.

Common Mistake

Area unit conversions

Do not say 1500 cm² is 15 m². Since 1 m² is 10,000 cm², 1500 cm² is only 0.15 m².

Exam technique

In the exam

Check whether you have been given radius or diameter before using any formula.
Write the formula first, then substitute the numbers carefully.
Keep calculator values unrounded until the final line, then round exactly as requested.

Self review

Check yourself

If a cylinder has diameter 18 cm, what radius should you use in the formula?
Which formula would you use for volume: $V=πr2hV=\pi r^2h$ or $TSA=2πr2+2πrh\text{TSA}=2\pi r^2+2\pi rh$ ?
In a painting question, why might you need to round the number of pots up?

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a solid cylinder

4. Half-cylinders

5. Working backwards

6. Units, paint and filling questions

Recap questions

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a solid cylinder

4. Half-cylinders

5. Working backwards

6. Units, paint and filling questions

Recap questions

Cylinders

What you'll learn

1. The circle facts you need first

2. Volume of a cylinder

3. Total surface area of a solid cylinder

4. Half-cylinders

5. Working backwards

6. Units, paint and filling questions

Recap questions

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 solid cylinder

4. Half-cylinders

5. Working backwards

6. Units, paint and filling questions

Recap questions

Cylinders Revision Guide