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Here we will learn about the surface area of a cone, including how to find the surface area given the radius and the slant height.

There are also cone* *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 cone** is the area which covers the outer surface of the cone.

For any cone, r is the** radius **of the** base of the cone**, l is the **slant height** of the cone, and h is the** perpendicular height** of the cone.

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

There is a **curved surface area** and a **circular base**. (The curved surface area of a cone is sometimes referred to as the lateral surface area of a cone).

The surface area of a cone is the area of the two parts added together,

\text{Surface area of a cone}=\pi{r}{l}+\pi{r^{2}}

The **perpendicular height** of a cone h is not required to calculate the surface area of a cone, although it is used to calculate the volume of the cone.

**Step-by-step guide:** __Volume of a cone__ (coming soon)

**Note:** We **cannot** use the formula for the surface area of a cone written above for an oblique cone. This is because the curved surface area does not have a constant slant height l from the **apex** (a **vertex**) to the circular base.

Calculate the surface area of the cone below to 1 decimal place.

Curved surface area =\pi{rl}=\pi\times{6}\times{8}=48\pi

Area of circular base =\pi{r^{2}}=\pi\times{6^{2}}=36\pi

We then add these together to find the total surface area of the cone,

Surface area of the cone = 48\pi + 36\pi = 84\pi = 263.9 cm^2 (1dp)

In order to calculate the surface area of a cone:

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

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

COMING SOONGet your free surface area of a cone worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONFind the curved surface area of the cone below, with radius 3cm and slant height 7cm . Write your answer to 1 decimal place.

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

We only need to calculate the area of the curved surface for this question.

\[\text{Curved surface area}=\pi rl\\
=\pi \times 3 \times 7\\
=21\pi \\\]

2**Add the areas together.**

We only need to find the curved surface area, so we can move on to step 3 .

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

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

21\pi=66.0 (1dp)The curved surface area of the cone is 66.0cm^2 (1dp) .

**Note:** If the question asks for 1 decimal place, the answer must have 1 decimal place, even if the digit is 0 .

Find the total surface area of the cone below to 1 decimal place.

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

\[\text{Curved surface area}=\pi rl\\
=\pi \times 6.8 \times 9.1\\
=\frac{1547}{25}\pi\\\]

\[\text{Area of circular base }=\pi r^2\\
=\pi \times 6.8^2\\
=\frac{1156}{25}\pi\\\]

**Add the areas together.**

The sum of the areas is

\frac{1547}{25}\pi+\frac{1156}{25}\pi=\frac{2703}{25}\pi .

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

We need to write the answer to 1 decimal place.

\frac{2703}{25}\pi=339.7\ (1dp)The total surface area of the cone is 339.7cm^2 (1dp) .

A chip shop sells mushy peas in a conical pot. The lid has a radius of 7cm , and the pot has a slant height of 10cm . Calculate the outside surface area of the closed pot, including the lid. Write your answer in terms of \pi .

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

\[\text{Curved surface area (the pot)}=\pi rl\\
=\pi \times 7 \times 10\\
= 70 \pi\]

\[\text{Area of a circle (the lid)}=\pi r^2\\
=\pi \times 7^2\\
=49 \pi\]

**Add the areas together.**

The sum of the areas is

70\pi+49\pi =119\pi .

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

We need to write the answer in terms of \pi , and state the units.

The total surface area of the cone is 119\pi cm^2 .

The roof of a tower is a solid cone. The radius of the circular base is 4.5m , and the roof has a slope length of 8.3m . Calculate the surface area of the roof. Write your answer in the form \frac{a}{b}\pi where a and b are integers with no common factors.

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

\[\text{Curved surface area}=\pi rl\\
=\pi \times 4.5 \times 8.3\\
=\frac{747}{20}\pi\\\]

\[\text{Area of circular base }=\pi r^2\\
=\pi \times 4.5^2\\
=\frac{81}{4}\pi\\\]

**Add the areas together.**

The sum of the areas is equal to

\frac{747}{20}\pi+\frac{81}{4}\pi=\frac{747}{20}\pi+\frac{405}{20}\pi=\frac{1152}{20}\pi .

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

We need to write the answer in the form \frac{a}{b}\pi where a and b are integers with no common factors. By simplifying the fraction fully, we get \frac{1152}{20}\pi=\frac{288}{5}\pi .

The surface area of the solid cone is \frac{288}{5}\pi\text{m}^{2} .

Calculate the surface area of a cone with a base diameter of 4cm and a slant height of 6cm . Give your answer to 3 significant figures.

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

We need to calculate the radius of the circular base first. As the diameter is twice the radius of a circle, we need to divide the diameter of the circle by 2 .

4\div2=2cmWe now have r=2cm .

\[\text{Curved surface area}=\pi rl\\
=\pi \times 2 \times 6\\
=12\pi\\\]

\[\text{Area of circular base }=\pi r^2\\
=\pi \times 2^2\\
=4\pi\\\]

**Add the areas together.**

Adding the two areas together, we get 12\pi + 4\pi = 16\pi .

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

We need to write the answer to 3 significant figures, and state the units.

16\pi=50.2\ (3sf)The surface area of the cone is 50.2cm^2 (3sf) .

A cone has a perpendicular height of 4cm , and a circular base with a radius of 3cm . The apex of the cone is directly above the centre of the circular base. Calculate the area of the curved surface in terms of \pi .

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

We need to know the slant height of the cone. Sketching a diagram of the cone with the dimensions given, we have

To find the slant height l , we need to use Pythagoras theorem.

a^2+b^2=c^2 . Given that a=4cm, b=3cm, and c=l , we have

\[4^2+3^2=l^2\\
16+9=l^2\\
l^2=25\\
l=5cm\]

Now we have the value for the slant height, we can find the curved surface area.

\[\text{Curved surface area}=\pi rl\\
=\pi \times 3 \times 5\\
=15\pi \]

**Add the areas together.**

We only need to calculate the area of the curved surface so we can move on to step 3 .

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

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

The curved surface area of the cone is 15\pi cm^2 .

**Using the correct formula**

There are several formulas that can be used, so we need to match the correct formula to the correct context.

**Rounding**

Do not round the answer until the end of the calculation. This will mean your final answer is more accurate.

**Using the radius or the diameter**

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

**Make sure you have the correct unit**- For area we use square units such as cm^2, m^2 .

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

1. Find the curved surface area of a cone of radius 10cm and slant height 12cm . Give your answer to 3 significant figures.

120 cm^2

376.991 cm^2

377 cm^2

3770 cm^2

We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.

\text{Curved surface area}=\pi rl\\ =\pi \times 12\times 10\\ =120\pi\\ =376.9911184…\\ =377 \ cm^2 \ \text{(3sf)}

2. A cone has a radius of 7.8mm and a slant height of 9.6mm . Calculate the curved surface area of the cone, to 3 significant figures.

74.9 mm^2

117mm^2

235 mm^2

235.242 mm^2

We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.

\text{Curved surface area}=\pi rl\\ =\pi \times 7.8\times 9.6\\ =235.2424579…\\ =235 \ \text{mm}^2 \ \text{(3sf)}

3. A candle is made in the shape of a cone. The radius of the base is 4cm , and the slant height is 7cm . Calculate the surface area of the candle to 1 decimal place.

138.2 cm^2

88.0cm^2

117.3cm^2

113.1cm^2

We are finding the total surface area of a cone so we find the curved surface area and add on the area of the circular base.

\text{Curved surface area}=\pi rl\\ =\pi \times 4 \times 7\\ = 28 \pi

\text{Area of circular base }=\pi r^2\\ =\pi \times 4^2\\ =16 \pi

The total surface area is 28\pi +16\pi =44\pi .

44\pi=138.2\text{cm}^{2}\ (1dp)

4. Find the total surface area of a cone of radius 6.5cm and slant height 9.3cm . Write your answer as a fraction in the form \frac{a}{b}\pi where a and b are integers. State the units in your answer.

\frac{1027}{10}\pi\text{cm}^{2}

\frac{5239}{40}\pi\text{cm}^{2}

\frac{17407}{40}\pi\text{cm}^{2}

\frac{1469}{20}\pi\text{cm}^{2}

We are finding the total surface area of a cone so we find the curved surface area and add on the area of the circular base.

\text{Curved surface area}=\pi rl\\ =\pi \times 6.5\times 9.3\\ =\frac{1209}{20}\pi

\text{Area of circular base }=\pi r^2\\ =\pi \times 6.5^2\\ =\frac{169}{4}\pi

The total surface area is \frac{1209}{20}\pi+\frac{169}{4}\pi=\frac{1207}{10}\pi .

Surface area is equal to \frac{1207}{10}\pi cm^2 .

5. A conical party hat has a diameter of 16cm , and a slant height of 11cm . An image is being placed on the entire outside surface of the party hat. What is the area of the image? Write your answer to in terms of \pi .

88\pi cm^2

176\pi cm^2

276 cm^2

\frac{704}{3}\pi\text{cm}^{2}

The diameter is twice the length of the radius and so r=16\div2=8cm . Substituting the values for r and l into the formula for the curved surface area we get,

\text{Curved surface area}=\pi rl\\ =\pi \times 8\times 11\\ =88\pi\\ =88\pi \ \text{cm}^2

6. A cone has a radius of 5cm and a perpendicular height of 12cm . The apex of the cone is directly above the centre of the circular base. Calculate the surface area of the cone to 3 significant figures.

94.2 cm^2

283 cm^2

188 cm^2

314 cm^2

Use Pythagoras theorem to calculate the slant height of the cone, then determine the surface area.

Pythagoras theorem is a^{2}+b^{2}=c^{2} .

a^{2}+b^{2}=c^{2}\\ 5^{2}+12^{2}=l^{2}\\ 25+144=l^{2}\\ l^{2}=169\\ l=\sqrt{169}\\ l=13\text{cm}

The slant height l=13cm .

\text{Curved surface area}=\pi rl\\ =\pi \times 5\times 13\\ =65\pi

\text{Area of circular base }=\pi r^2\\ =\pi \times 5^2\\ =25\pi

The total surface area is 65\pi+25\pi=90\pi=283\ \text{(3sf)} .

The total surface area of the cone is 283cm^2 (3sf) .

\text{Curved surface area of a cone}=\pi rl

1. Below is a diagram of a cone.

Calculate the curved surface area of the cone.

Write your answer to 3 significant figures.

**(2 marks)**

Show answer

= \pi \times 12.6 \times 15.3

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2. Here is a cone.

Calculate the total surface area of the cone.

Write your answer in terms of \pi .

**(3 marks)**

Show answer

= \pi \times 13.9 \times 16.1=223.79\pi

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3. A cone has a diameter of 6cm and a slant height of 7cm . Calculate the total surface area of the cone. Write your answer to 1 decimal place.

**(5 marks)**

Show answer

r=6\div{2}=3\text{cm}

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4. A cone has a circular base with an area of 25\pi and a slant height of 6cm . Calculate the total surface area of the cone in terms of \pi .

**(3 marks)**

Show answer

r=\sqrt{25}=5\text{cm}

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You have now learned how to:

- Calculate surface areas of cones

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