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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 \; c^m2 (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 283 \; 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.

**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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