Volume of a Cone

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Volume of a Cone

Just like the familiar cylinder, cones are everywhere in our daily lives, from ice cream cones to traffic cones. Learning the volume of a cone is not just an academic exercise but a practical skill that can be applied in various real-world scenarios.

The volume of a cone is the amount of space inside a cone. This volume is calculated using the formula $V=\frac{1}{3} \pi r^{2}h$, where $V$ stands for volume, $r$ is the radius of the cone's base and $h$ is its perpendicular height.

Understanding the Volume of a Cone

The key to understanding the volume of a cone is to realise that it is a third of the volume of a cylinder with the same base and height. This concept is crucial in grasping why the formula includes the $\frac{1}{3}$ factor.

Volume of a Cone – Cubic Units

Determining the volume of 3D shapes involves using specific formulas, and it's equally important to label the final results with the appropriate units.

Choosing the Correct Volume Units

Volume in cones, like all three-dimensional objects, is expressed in cubic units. These units depend on the measurement of radius and height. For instance, if these measurements are in centimetres, the volume will be in cubic centimetres (cm³).

Volume of a Cone – Step-by-Step Instructions

Here’s how to calculate the volume of a cone, step by step:

Step Number Directions Example
1 Identify the radius and perpendicular height of the cone. Radius $r = 3$ cm, Height $h = 12$ cm
2 Substitute the values into the formula $V=\frac{1}{3} \pi r^{2}h$. $V=\frac{1}{3} \pi \times 3^2 \times 12$
3 Calculate the volume, attending to rounding rules. $V=\frac{1}{3} \pi \times 9 \times 12 = 36\pi$ cm³ approx. $113.1$ cm³ to 1 decimal place.
4 Write the final answer with the correct units. Volume of the cone is $113.1$ cm³ to 1 decimal place.

Let’s work through an example of calculating the volume of a cone.

Imagine a party hat shaped like a cone with a radius of $4$ in. and a height of $10$ in.

Use the formula $V=\frac{1}{3} \pi r^{2}h$ to calculate the volume.
Calculate the volume using pi and round to the nearest tenth.

In addition to cones, you can determine the volume of prisms by employing different formulas while following a parallel approach.

Calculating Volume 'In Terms of $\pi$'

Calculating volume 'in terms of $\pi$' means leaving $\pi$ as is, without converting it to a decimal. This method is more precise and often used for exactness in mathematical and scientific calculations.

Let’s practise calculating the volume of a cone and leaving the answer in terms of $\pi$.

Use the formula $V=\frac{1}{3} \pi r^{2}h$ to calculate the volume.
Calculate the volume leaving the answer in terms of $\pi$.

Volume of a Cone – Real-World Problems

Cones are everywhere in our daily life, not just in maths books. Think of a party hat or a funnel in a kitchen. Knowing how to calculate their volume can be really handy in real-world situations.

Let's explore real-world problems involving cones.

An ice cream cone has a radius of $2.5$ cm and a height of $6$ cm. How much ice cream can it hold?
A conical paper cup has a radius of 7 cm and a height of 24 cm. What is its volume?

Volume of a Cone – Exercises

Now that you have learnt the formula and process for calculating the volume of a cone, try these exercises!

A cone has a radius of 5 cm and a height of 10 cm. Calculate its volume, rounding to the nearest tenth.
Find the volume of a cone with a radius of 4 inches and a height of 12 inches, rounding the answer to the nearest tenth.
A cone has a radius of 3 m and a height of 9 m. What is its volume, rounded to the nearest tenth?
Calculate the volume of a cone in terms of $\pi$ if it has a radius of 8 cm and a height of 20 cm.

Volume of a Cone – Summary

Key Points from this Text:

• The formula for calculating the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.
• To find the volume, identify the radius and perpendicular height of the cone.
• Substitute the values into the formula and use a calculator to compute, rounding to the nearest tenth or leaving in terms of $\pi$.
• Real-world examples of cones include ice cream cones and traffic cones, making this concept widely applicable.

For more on volume, have a look at Volume of a Sphere.

Volume of a Cone – Frequently Asked Questions

What is the formula for the volume of a cone?
How do I find the radius and height of a cone?
Can I calculate the volume of a cone using its diameter?
Can I calculate the volume of a cone using the sloped height?
Why is the cone's volume one-third that of a cylinder?
What units should I use for measuring the volume of a cone?
How do I calculate the volume of a cone in terms of pi?
Can I use a calculator to find the volume of a cone?
Is it necessary to round the volume of a cone?
What are some real-world examples of cones?
How do I convert volume units, like from cm³ to m³?

Volume of a Cone exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the video Volume of a Cone.
• Understand what the volume of a cone measures.

Hints

A cone is a three-dimensional shape, such as this party hat! Some other common cones are:

• ice cream cones
• traffic cones
• funnels

Like any three-dimensional shape, the volume measures the interior 3-D space.

Solution

The volume of a cone measures the amount of space inside the cone.

• Using a formula to find the volume of a cone.

Hints

The radius is the distance from the centre of a circle to the outside.

The height of a cone measures the distance from the centre of the circle to the tip of the cone. This line of measurement must be perpendicular to the base.

Substituting a value for a variable means replacing the variable with a specific number in an equation or expression, which allows you to solve or simplify it.

For example, if we know that $x=4$, and we need to evaluate $x - 6$, we would then substitute to write $\bf{4} - 6$.

Solution

The radius of the cone is $\bf{2\:cm}$.

The height of the cone is $\bf{4\:cm}$.

The formula used to find the volume of the cone is $\bf{V=\frac{1}{3} \pi r^2h}$.

Once the values for the radius and height are identified, substitute them for the appropriate variables, and write the equation: $\bf{V=\frac{1}{3} \pi (2^2)(4)}$.

• Volume of a cone.

Hints

To use a formula, substitute in the values for the corresponding variable.

For example, the cone seen here has the following measurements:

$r=3\:cm$

$h=2\:cm$

The formula to find the volume is: $V=\frac{1}{3}\pi\: r^2h$

When the values are substituted, the equation is: $V=\frac{1}{3}\pi \:(3^2)2$.

Solution

Substitute the values of the radius and height into the formula, $V=\frac{1}{3}\pi \:r^2h$, for $r$ and $h$.

• Find the volume of a cone.

Hints

The formula for volume of a cone is $V=\frac{1}{3}\pi r^2h$.

In terms of $\pi$ means that you don't calculate $\pi$. Leave it as the symbol in the answer instead.

If we were calculating the volume of this cone (in terms of $\pi$), first we would substitute in the known values.

$V=\frac{1}{3} \pi \:r^2h$

$V=\frac{1}{3} \pi \:(6^2)(15)$

Then we would calculate the product of all the numbers except pi.

$\frac{1}{3}(6^2)(15)=180$

Finally, the pi symbol can be written next to the product, to show the exact value for the volume.

$V=180\pi\:m^3$

Solution

$r=2$

$h=6$

$V=\frac{1}{3}\pi r^2 h$

$V=\frac{1}{3} \pi (2^2) (6)$

$\frac{1}{3}(2^2)(6)=8$

$V=8\pi\:in^3$

• Identify the variables in the formula for the volume of a cone.

Hints

Remember, this problem is about the volume of a cone. Which measurements are important to finding this value?

Look at the cone. What measurements do the $h$ and $r$ represent?

Solution

The variables and symbols in the Volume of a Cone formula are:

V = volume

h = height

$\pi$ = pi

• Calculate the volume of a cone using the formula.

Hints

Formula for the volume of a cone: $V=\frac{1}{3}\pi r ^2 h$

If you do not have a calculator at hand, you can use the approximate value of pi, $3.14...$.

Remember, the $r$ is the radius of the cone, and the $h$ is the height of the cone.

Make sure to take your time and show your work to find the volume.

• Identify the radius and height.
• Substitute these values in the formula $V=\frac{1}{3}\pi r^2 h$.
• Calculate the volume and round to one decimal place.
Solution

The volume for each cone:

V = $804.2\:cm^3$

V = $212.1\:cm^3$

V = $83.8\:cm^3$