### Volume of a Cone

#### Lesson Objective

In this lesson, we will learn about the volume of a cone.

#### About This Lesson

In this lesson, we will:

- Learn about the formula for the volume of a cone and see how it is related to the volume of a cylinder.
- Look at an example on using the formula to calculate the cone's volume
- Look at another example on using the formula to calculate the base radius of a cone

The study tips and math video below will explain more.

### Study Tips

#### Tip #1

In the previous lesson, we learned that the volume of a cylinder is πr^{2}h. the volume of a cone is one third of the cylinder's volume.

Hence, for a cone with height h and base radius r, the volume, V of the cone will be:

The math video below will give more explanation on this formula. Also, we will see some examples on how to use it.

### Math Video

#### Lesson Video

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00:00:03.140

In this lesson, we will learn about the volume of a cone.

00:00:07.230

Consider a cylinder with the height h, and base radius r.

00:00:13.120

Now, we already know that the volume of a cylinder, V, is pi r square h.

00:00:20.120

With this, let's change this cylinder into a cone.

00:00:26.040

Now, we have a cone with the height h, and base radius r.

00:00:31.120

Base on calculations, the volume of the cone found to be one third of a volume of a cylinder.

00:00:38.000

Therefore, the volume of a cone, V, is one third pi r square h.

00:00:45.090

Let's see some examples on using this formula. For these examples, we take pi as 3.14.

00:00:55.090

Find the volume of this cone when its height is 5cm, and its base radius is 3cm.

00:01:02.040

Now, we start with the formula for the volume of a cone, V equal 1 third pi r square h.

00:01:09.180

Since the radius is 3cm, we can substitute r with 3.

00:01:15.220

Now, let's simplify 3 square. 3 square is the same as 3 multiply by 3. This gives 9.

00:01:26.080

Let's put this back here.

00:01:29.140

Next, since the height is given as 5 cm, we can substitute h with 5.

00:01:36.220

We can simplify by multiplying 9 with 5. This gives 45.

00:01:43.160

Next, since pi is given as 3.14, we can substitute this pi with 3.14.

00:01:52.020

3.14 multiply by 45 gives 141.30.

00:01:59.020

Now, this term is the same as, 1 bracket 141.30, over 3.

00:02:07.020

1 multiply by 141.30 gives back 141.30.

00:02:14.120

141.30 divides by 3, gives 47.10.

00:02:21.150

Now, this number has no meaning unless we include the units for it.

00:02:26.130

Since the units are given in centimeter, the volume will be in cubic centimeter.

00:02:32.160

There, the volume of this cone is 47.10 cubic centimeter.

00:02:40.120

Next example, the volume of this cone is 12.56 cubic feet, and its height is 3cm. Find its radius, r.

00:02:51.180

To solve this, we start with the formula for the volume of a cone, V equals 1 third pi r square h.

00:03:00.010

Here, we can see that since the volume, pi, and h are given, we can find r, by solving this equation for r. Here’s how.

00:03:11.140

First, it is easier to work with this equation if we write 1 third pi r square h in the form of fraction. By doing so, we have, 1 pi r square h, over 3.

00:03:25.040

1 pi r square h is the same as, pi r square h.

00:03:31:040

Next, note that we can remove the fraction in this equation, by multiplying both sides of the equation with 3.

00:03:39.160

This gives, 3 'V' equals to pi r square h.

00:03:45.130

Now, the volume of the cone is 12.56 ft. Hence, we can substitute V with 12.56.

00:03:53.220

3 multiply by 12.56 gives 37.68

00:04:00.020

Next, since the height of the cone is 3 ft, we can substitute h with 3.

00:04:07.020

Similarly, since pi is given as 3.14, we can substitute this pi with 3.14.

00:04:15.180

We can simplify by multiplying 3.14 with 3. This gives 9.42.

00:04:24.040

Alright, now we have 9.42 r square, equals to, 37.68.

00:04:32.130

Let's rewrite this, so it will look neater.

00:04:36.230

To get closer to finding r, we need to remove 9.42.

00:04:43.090

We can do so, by dividing both sides of the equation with 9.42. This gives, r square equals to 37.68 over 9.42.

00:04:57.080

37.68 divides by 9.42, gives 4.

00:05:04.010

Now, since r square is equals to 4, we can find r, by calculating the square root of 4.

00:05:13.040

Square root of 4 gives 2.

00:05:16.240

Again, this number has no meaning unless we include the unit for it.

00:05:22.050

Since the volume is in cubic feet, the radius of the base of this cone will be in feet.

00:05:28.170

Hence, the radius of the base of this cone is 2 ft.

00:05:35.180

This is all for this lesson. Try out the practice question to further your understanding.

### Practice Questions & More

#### Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on the volume of a cone or pick your choice of question below.

- Question 1 on finding the volume of a cone
- Question 2 on finding the base radius of a cone

#### Site-Search and Q&A Library

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