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 learn that the volume of a cylinder is πr²h. Now, 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:

a cone with a height h and base radius r

where π is a number that is approximately equals to 3.14.

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

Math Video

Lesson Video

If the above player doesn't work, try this direct link.

Math Video Transcript

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

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