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Applying formulae to calculate and solve problems involving

Area of quadrilaterals Rounding numbersThis topic is relevant for:

Here we will learn about the surface area of a cylinder, including how to calculate the curved surface area of a cylinder given its radius and its perpendicular height. We will also look at calculating the total surface area of a cone.

There are also cylinder worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

The **surface area of a cylinder** is the area which covers the outer surface of a cylinder.

In order to calculate the total surface area of a cylinder we need to find the area of the three parts of the surface of the cylinder and add them together.

There is a **curved surface area** and** two circular bases** (the curved surface area of a cylinder is sometimes referred to as the lateral surface area of a cylinder).

r is the radius of the cylinder and h is the **perpendicular height** of a cylinder.

The **curved surface area** of a cylinder is actually a **rectangle**.

The **circumference of the circle **is the** length of the rectangle**.

The **height ** h ** of the cylinder** is the **height of the rectangle**.

To work out the** formula for the surface area of a cylinder** we need to work out the curved surface area and the area of the two circular bases

The curved surface area of a cylinder is a rectangular shape. The base of the rectangle is the circumference of the circle:

\text{Circumference of a circle}=\pi d=2\pi rThe height of the rectangle is height of the cylinder given by h.

To find the area of a rectangle we need to multiply the base 2\pi r by the height h, this gives us:

Formula for the **curved surface area**:

Then we need to calculate the area of the base of the cylinder, this is a circle.

Formula for the** area of circle**:

The** area of the top** and the **area of the base** of a cylinder are the **same**.

So to find the the **total surface area**, we can add the curved surface area to the area of the two circles;

E.g.

Find the total surface area of this cylinder with radius of the base 7 cm and perpendicular height 10 cm.

First, we need to find the curved surface area of the cylinder:

\text{Curved surface area}=2\pi rh=2\times \pi \times 7 \times 10=140\piWe then need to find the area of the base of the cylinder:

\text{Area of a circle}=\pi r^2=\pi \times 7^2=49\piThe area of the top of the cylinder is the same as the area of the base so,

**Total surface area:**

Since the measurements of the cylinder are in cm, the surface area will be measured in cm^2.

The surface area of the cylinder is 747.7 \; cm^2 \; (1dp)

In order to calculate the surface area of a cylinder:

**Work out the area of each face.****Add the areas together.****Write the answer, including the units.**

Get your free surface area of a cylinder worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free surface area of a cylinder worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEFind the curved surface area of the cylinder below, with radius 3 \; cm and perpendicular height 8 \; cm.

Give your answer to 1 decimal place.

**Work out the area of each face.**

This question is only asking us for the curved surface area so we only need to find the area of one face.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 3 \times 8\\\\ &=150.7964474… \end{aligned}2**Add the areas together.**

Since we are only finding the curved surface area, we do not need to add any other areas.

3**Write the answer, including the units.**

We need to round the answer to 1 decimal place and include the units.

The curved surface area of the cylinder is: 150.8 \; cm^2 (to 1 dp )

Find the curved surface area of the cylinder below, with radius 5 \; cm and perpendicular height 8 \;cm.

Leave your answer in terms of \pi .

**Work out the area of each face.**

This question is only asking us for the curved surface area so we only need to find the area of one face.

**Add the areas together.**

Since we are only finding the curved surface area, we do not need to add any other areas.

**Write the answer, including the units.**

We need to leave the answer in terms of \pi and include the units.

\text{Curved surface area }=80 \pi \mathrm{cm}^{2}

Find the total surface area of the cylinder below, with radius 4 \; cm and perpendicular height 9 \;cm.

Give your answer to 3 significant figures.

**Work out the area of each face.**

This question is asking us for the total surface area of the cylinder. We need to find the area of each face and add them together.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 4 \times 9\\\\ &=226.1946… \end{aligned}

The area of the base is

\pi \times 4^{2}=50.2654...

The top face is the same as the bottom face so the area of the top is also 50.2654...

**Add the areas together.**

The sum of the areas is 226.1946+50.2654+50.2654 = 326.7254

**Write the answer, including the units.**

We need to round to 3 significant figures and include units.

The total surface area of the cylinder is: 327 \; cm^2 (to \; 3 sf)

Find the total surface area of the cylinder below, with radius 3.5 \; cm and perpendicular height 9.4 \;cm.

Give your answer to 3 significant figures.

**Work out the area of each face.**

This question is asking us for the total surface area of the cylinder. We need to find the area of each face and add them together.

\begin{aligned}\\ \text{Area of base }&= \pi r^{2}\\\\ &=\pi \times 3.5^{2}\\\\ &=38.4845 \end{aligned}

Area of top: 38.4845

**Add the areas together.**

The sum of the areas is 206.7167+38.4845+38.4845=283.6857

**Write the answer, including the units.**

We need to round to 3 significant figures and include units

The total surface area of the cylinder is 284 \; cm^2 (to \; 3 sf)

Find the total surface area of the cylinder below, with radius 6 \; cm and perpendicular height 8 \;cm.

Leave your answer in terms of \pi .

**Work out the area of each face.**

This question is asking us for the answer in terms of \pi . When calculating the area of each face we need to leave the answers in terms of \pi .

\begin{aligned} \\ \text{Area of base }&= \pi r^{2}\\\\ &=\pi \times 6^{2}\\\\ &=36 \pi \end{aligned}

Area of top: 36 \pi

**Add the areas together.**

\text{The sum of the areas is }96 \pi +36 \pi + 36 \pi = 168 \pi

**Write the answer, including the units.**

We need to leave the answer in terms of \pi and include the units.

The total surface area of the cylinder is:

168 \pi \; \mathrm{cm}^{2}

Find the total surface area of the cylinder below, with radius 2.7 \; cm and perpendicular height 6.3 \;cm.

Leave your answer in terms of \pi .

**Work out the area of each face.**

This question is asking us for the answer in terms of \pi . When calculating the area of each face we need to leave the answers in terms of \pi .

\begin{aligned} \\ \text{Area of base }&= \pi r^{2}\\\\ &=\pi \times 2.7^{2}\\\\ &=7.29 \pi \end{aligned}

Area of top: 7.29 \pi

**Add the areas together.**

\text{The sum of the areas is }34.02 \pi +7.29 \pi + 7.29 \pi = 48.6 \pi

**Write the answer, including the units.**

We need to leave the answer in terms of \pi and include the units.

The total surface area of the cylinder is:

48.6 \pi \; \mathrm{cm}^{2}

**Rounding**

It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate.

r**Using the radius or the diamete**

It is a common error to mix up radius and diameter. Remember the radius is half of the diameter and the diameter is double the radius.

**Correct units**

For area we use square units such as cm^2.

For volume we use cube units such as cm^3.

**Check what you have answered the question**

Double check if you are asked to just find the curved surface area or the total surface area.

Surface area of a cylinder is part of our series of lessons to support revision on cylinders. You may find it helpful to start with the main cylinder lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. Find the curved surface area of a cylinder of radius 5.2 \; cm and perpendicular height 8.3 \; cm .

Give your answer to 3 significant figures.

272 \; cm^2

273 \; cm^2

271 \; cm^2

274 \; cm^2

We are finding the curved surface area of a cylinder so we substitute the value of

r and h into the formula.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 5.2\times 8.3\\\\ &=271.182…\\\\ &=271 \ cm^2 \ \text{(to 3 sf)} \end{aligned}

2. Find the curved surface area of a cylinder of radius 6.7 \; cm and perpendicular height 4.9 \; cm .

Give your answer to 3 significant figures.

690 \; cm^2

659 \; cm^2

692 \;cm^2

691 \; cm^2

We are finding the curved surface area of a cylinder so we substitute the value of

r and h into the formula.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 6.7\times 4.9\\\\ &=691.027…\\\\ &=691 \ cm^2 \ \text{(to 3 sf)} \end{aligned}

3. Find the curved surface area of a cylinder of radius 2\; cm and perpendicular height 7 \; cm .

Leave your answer to in terms of \pi .

22\pi \; cm^2

28\pi \; cm^2

24\pi \;cm^2

26\pi \; cm^2

We are finding the curved surface area of a cylinder so we substitute the value of

r and h into the formula.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 2\times 7\\\\ &=28\pi\\\\ &=28\pi \ cm^2 \end{aligned}

4. Find the total surface area of a cylinder of radius 6.1\; cm and perpendicular height 3.9 \; cm .

Give your answer to 3 significant figures.

384 \; cm^2

254 \; cm^2

255 \;cm^2

383 \; cm^2

We are finding the total surface area of a cylinder so we find the area of each face and add them together.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\\\ &=2 \times \pi \times 6.1\times 3.9\\\\ &=149.4769… \end{aligned}

\begin{aligned} \\ \text{Area of circle }&=\pi r^2\\\\ &=\pi \times 6.1^2\\\\ &=116.8986… \end{aligned}

Total surface area: 149.4769+116.8986+116.8986=383.2741

Surface area = 383 \; cm^2 \; (3sf)

5. Find the total surface area of a cylinder of radius 2.7\; cm and perpendicular height 8.6 \; cm .

Give your answer to 3 significant figures.

192 \; cm^2

191 \; cm^2

610 \;cm^2

611 \; cm^2

We are finding the total surface area of a cylinder so we find the area of each face and add them together.

\begin{aligned} \text{Curved surface area}&=2\pi rh\\ &=2 \times \pi \times 2.7\times 8.6\\ &=145.8955… \end{aligned}

\begin{aligned} \\ \text{Area of circle }&=\pi r^2\\ &=\pi \times 2.7^2\\ &=22.9022… \end{aligned}

Total surface area: 145.8955+22.9022+22.9022=191.6999

Surface area = 192 \; cm^2 \; (3sf)

6. Find the total surface area of a cylinder of radius 2\; cm and perpendicular height 6 \; cm .

Leave your answer to in terms of \pi .

34\pi \; cm^2

38\pi \; cm^2

38\pi \;cm^2

32\pi \; cm^2

We are finding the total surface area of a cylinder so we find the area of each face and add them together

\begin{aligned} \text{Curved surface area}&=2\pi rh\\ &=2 \times \pi \times 2\times 6\\ &=24\pi \end{aligned}

\begin{aligned} \\ \text{Area of circle }&=\pi r^2\\ &=\pi \times 2^2\\ &=4 \pi \end{aligned}

Total surface area: 24\pi + 4 \pi + 4 \pi = 32\pi \mathrm{cm}^{2}

1. Here is a cylinder.

Calculate the total surface area of the cylinder.

Give your answer to 3 significant figures.

**(3 marks)**

Show answer

2\times \pi \times 5.1 \times 4.3=137.790…

**(1)**

137.790… + 2\times \pi \times 5.1^2=137.790…+163.425…

**(1)**

301.21…=301

**(1)**

2. Here is a cylinder.

Calculate the total surface area of the cylinder.

Leave your answer in terms of \pi .

**(3 marks)**

Show answer

2\times \pi \times 3 \times 7=42\pi

**(1)**

42\pi + 2\times \pi \times 3^2=42\pi+18\pi

**(1)**

60\pi

**(1)**

3. Sarah has to cover 4 containers completely with paint.

Each container is the shape of a cylinder with a top and a bottom.

The container had a radius of 0.7 \; m and a height of 1.3\; m.

Sarah has 8 tins of paint.

Each tin of paint covers 4 \; m^2 .

Does Sarah have enough paint to cover the containers?

You must show how you get your answer.

**(5 marks)**

Show answer

\pi \times 0.7^2=1.5393…

For finding the area of a circle

**(1)**

2\times \pi \times 0.7 \times 1.3=5.7176

For finding the curved surface area of a cylinder

**(1)**

4\times(5.7176…+2\times 1.5393…)=35.18…

For finding the Ttotal surface area of 4 cylinders

**(1)**

8\times 4=32

For finding the coverage of the tins

**(1)**

**No**, because 32 is less than 35.18…

For the final answer with supporting workings

**(1)**

You have now learned how to:

- Work out the volume of a cylinder
- Solve problems involving the volume of a cylinder

For GCSE we look at right circular cylinders – where the bases are parallel planes and the height is perpendicular to these bases. It is possible to have oblique cylinders.

It is also possible to have a cylinder with an ellipse as its base.

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#### GCSE Maths Papers - November 2022 Topics

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Practice paper packs based on the November advanced information for Edexcel 2022 Foundation and Higher exams.

Designed to help your GCSE students revise some of the topics that are likely to come up in November exams.