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Here we will learn about cones, including how to find the volume of a cone and how to find the surface area of a cone.
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.
A cone is a three dimensional object with a circular base that tapers to a point (a vertex) that is directly above the centre of the base.
This type of cone is sometimes referred to as a “right circular cone”.
The radius of the circular base of a cone is r
The perpendicular height of a cone is h
The slant height of a cone is l
The volume of a cone is how much space there is inside a cone.
The formula for the volume of a cone is:
\text{Volume}=\frac{1}{3} \pi r^2 hE.g. Find the volume of the cone
Step by step guide: Volume of a cone (coming soon)
In order to calculate the volume of a cone:
Get your free cone worksheet of 20+ questions and answers. Includes reasoning and applied questions.
COMING SOONGet your free cone worksheet of 20+ questions and answers. Includes reasoning and applied questions.
COMING SOONFind the volume of the cone with radius 5.3 \ cm and perpendicular height 7.8 \ cm .
Give your answer to 3 significant figures.
2Substitute the given values.
\begin{aligned} \text { Volume } &=\frac{1}{3} \pi r^{2} h \\\\ &=\frac{1}{3} \times \pi \times 5.3^{2} \times 7.8 \end{aligned}3Work out the calculation.
\begin{aligned} &=\frac{1}{3} \times \pi \times 5.3^{2} \times 7.8 \\\\ &=229.443 \ldots \end{aligned}4Write the final answer, including the units.
=229 \ cm^3 \ (to \ 3.s.f)Find the volume of the cone with radius 9 \ cm and perpendicular height 11 \ cm .
Leave your answer in terms of \pi .
Write down the formula.
Substitute the given values.
Work out the calculation.
Write the final answer, including the units.
The question asks for the answer in terms of \pi so the final answer is =297\pi \ cm^3
The surface area of a cone is the area which covers the outer surface of a cone.
The surface area is made up of two parts, a curved surface area and a circular base.
The formula for calculating the curved surface area of a cone is:
\text{Curved surface area}=\pi rlThe formula for calculating the area of a circle:
\text{Area of circle}=\pi r^2For the total surface area, we can add the two parts together:
\text{total surface area}=\pi rl+\pi r^2E.g.
In order to calculate the surface area of a cone:
Find the curved surface area of the cone with radius 4.3 \ cm and slant height 9.6 \ cm.
Give your answer to 3 significant figures.
Work out the area of each face
\begin{aligned}
\text{Area of circle }&=\pi r^2\\\\
&=\pi \times 4.3^2\\\\
&=58.0880…
\end{aligned}
Add the areas together.
Total surface area: 129.6849+58.0880=187.7729...
Include units.
Surface area =188cm^2 \ (3sf)
Find the curved surface area of the cone with radius 8 \ cm and slant height 13 \ cm.
Leave your answer in terms of \pi .
Work out the area of each face
\begin{aligned}
\text{Area of circle }&=\pi r^2\\\\
&=\pi \times 8^2\\\\
&=64\pi
\end{aligned}
Add the areas together.
Total surface area: 104\pi +64\pi = 168\pi
Include units.
There are several formulas that can be used, so we need to match the correct formula to the correct context
It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate.
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.
For area we use square units such as cm^2.
For volume we use cube units such as cm^3.
1. Find the volume of a cone of radius 9.4 \ cm and perpendicular height 8.7 \ cm
Give your answer to 3 significant figures.
We are finding the volume of a cone so we substitute the value of r and h into the formula.
\begin{aligned} V&=\frac{1}{3} \pi r^2 h\\\\ V&=\frac{1}{3}\times \pi \times 9.4^2 \times 8.7\\\\ V&=805.014…\\\\ V&=805 \ cm^3 \ \text{(to 3 sf)} \end{aligned}
2. Find the volume of a cone of radius 8 \ cm and perpendicular height 6 \ cm
Leave your answer in terms of \pi .
We are finding the volume of a cone so we substitute the value of r and h into the formula.
\begin{aligned} V&=\frac{1}{3} \pi r^2 h\\\\ V&=\frac{1}{3}\times \pi \times 8^2 \times 6\\\\ V&=128\pi\\\\ V&=128\pi \ cm^3 \end{aligned}
3. Find the curved surface area of a cone of radius 4.3 \ cm and slant height 6.2 \ cm.
Give your answer to 1 decimal place.
We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.
\begin{aligned} \text{Curved surface area}&=\pi rl\\\\ &=\pi \times 4.3\times 6.2\\\\ &=83.754…\\\\ &=83.8 \ cm^2 \ \text{(to 1 dp)} \end{aligned}
4. Find the curved surface area of a cone of radius 7 \ cm and slant height 9 \ cm.
Leave your answer in terms of \pi .
We are finding the curved surface area of a cone so we substitute the value of r and l into the formula.
\begin{aligned} \text{Curved surface area}&=\pi rl\\\\ &=\pi \times 7\times 9\\\\ &=63\pi\\\\ &=63\pi \ cm^2 \end{aligned}
5. Find the total surface area of a cone of radius 5.9 \ cm and slant height 8.5 \ cm.
Give your answer to 3 significant figures.
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.
\begin{aligned} \text{TOTAL surface area}&=\pi rl+\pi r^2\\\\ &=\pi \times 5.9\times 8.5 + \pi \times 5.9^2\\\\ &=157.5508… + 109.3588…\\\\ &=266.90…\\\\ &=267 \ cm^2 \ \text{(to 3 sf)} \end{aligned}
6. Find the total surface area of a cone of radius 7 \ cm and slant height 10 \ cm.
Leave your answer in terms of \pi.
Make sure you find the curved surface area and the area of the circular base
\begin{aligned} \text{total surface area}&=\pi rl+\pi r^2\\\\ &=\pi \times 7\times 10 + \pi \times 7^2\\\\ &=70\pi + 49\pi\\\\ &=119\pi\\\\ &=119\pi \ cm^2 \end{aligned}
1. Here is a cone with radius 7.3 \ cm and perpendicular height 9.5 \ cm.
Find the volume of the cone.
Give your answer to 3 significant figures.
(2 marks)
(1)
530.148…=530(1)
2. Here is a cone.
Find the total surface area of the cone.
Give your answer to 3 significant figures.
(3 marks)
(1)
835.663…+\pi \times 14^2(1)
1451.4…=1450(1)
3. A container is a cone of radius 60 \ cm and perpendicular height 80 \ cm.
Water fills the container at a rate of 9000 \ cm^3 per minute.
How long does it take to fill the container?
Give your answer to the nearest minute.
(3 marks)
(1)
301 592.89… \div 3000=33.51…(1)
33.51…= 34 \ \text{minutes}(1)
4. Here is a cone.
Find the volume of the cone..
Give your answer to 3 significant figures.
(3 marks)
(1)
V=\frac{1}{3} \times \pi \times 4^2 \times 9.1651…(1)
153.563…=154(1)
5. A cone has a radius of 11 \ cm.
It has a volume of 2000 \ cm^3.
Find the total surface area of the cone.
Give your answer to 3 significant figures.
(5 marks)
(1)
l=\sqrt{11^2 + 15.783…^2}=19.2388…(1)
CSA= \pi \times 11\times 19.2388…=664.847…(1)
664.847…+\pi \times 11^2(1)
1044.97…=1040(1)
You have now learned how to:
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