In this explainer, we will learn how to calculate volumes of cones and solve problems including real-life situations.

While there exist more general definitions of cones, in this explainer, we will only be considering right circular cones. Let us recall precisely what this means with a mathematical definition.

### Definition: Right Circular Cones

A right circular cone is a three-dimensional shape with a circular base whose apex (or vertex) lies directly above the center of the base.

Its height is the distance from the apex to the base.

Its radius is the radius of the circular base.

Its slant height is the distance from the apex to any point lying on the circumference of the base.

Since we will only be considering right circular cones in this explainer, from now on, we will simply refer to them as cones. Let us now consider how we might find the volume of a cone.

Imagine that we can fill a cone completely with, say, water. If we pour this water into a cylinder of the same base and height as the cone, we will observe that the level of water is exactly at one-third of the height of the cylinder.

Now, we know that the volume of a cylinder is given by the formula where is the radius of the cylinder and is the height. If we take one-third of this, we arrive at the following formula for the volume of a cone.

### Formula: Volume of a Cone

The volume of a cone is equal to where is the radius of the base and is the height.

Thus, in any situation where we need to calculate the volume of a cone, we can do this by identifying the radius and height and substituting their values into the above formula. Let us consider an example where we use the formula for the volume of a cone given a diagram.

### Example 1: Finding the Volume of a Cone

Determine the volume of the right circular cone in terms of .

### Answer

The volume of a right circular cone is given by the formula where is the radius of the base and is the perpendicular distance from the apex to the base. In the diagram, we can see that the radius is 20 cm and the height is 24 cm. Substituting this into the formula, we get

In the previous example, we had the ideal scenario: we were given the radius and height, and we could use these to directly calculate the volume. Often, we will have situations where we are not explicitly given the measurements we need but have enough information to find the necessary values by deduction. In particular, recall that a right circular cone has its apex directly above the center of the circular base. This means that the slant height forms a right triangle with the height and radius as shown below.

Using the Pythagorean theorem, if we are given two of the measurements above, we can find the third. For instance, suppose the slant height is and the radius is and we want to find the height . Then, we have

We can rearrange this to write it in terms of as follows:

We can then put this value for , along with the given value for , into the formula for the volume. Let us consider an instance of this in the next example.

Example 2: Finding the Volume of a Cone given Its Height and Slant Height

Determine the volume of the given right circular cone in terms of .

### Answer

To find the volume of a right circular cone, recall that we have the formula where is the radius of the base and is the height (i.e., the distance from the apex to the base). We have been given the height, which is 48 cm, and the slant height, which is 60 cm, but not the radius.

To find the radius, we can make use of the fact that the slant height, height, and radius form a right triangle, as shown below.

Therefore, we can use the Pythagorean theorem to find :

Taking the square root of both sides, we find

We can now substitute this value, and , into the formula for the volume to get

Another type of problem involves the opposite question: if we are given the volume of a cone, can we find the other measurements? The answer to this is that as long as we can determine two out of the three unknowns (volume, height, and radius), we can use the formula for the volume of a cone to find the third by rearrangement. Let us now consider an example of this.

Example 3: Finding the Base Diameter of a Cone given Its Volume and Height

The volume of a cone is cm3, and its height is 12 cm. Find its diameter.

### Answer

Since we have been given the volume of a cone and its height, we recall the formula that relates the two, along with the radius: where is the volume, is the radius of the base, and is the height. Although this formula does not explicitly include the diameter, since the diameter is twice the radius, we can first concentrate on finding the radius.

Substituting and into the formula, we have

We can rearrange this to get . First, we divide both sides by :

Then, we can multiply both sides by 3, and divide both sides by 12 to get

Next, to find , we take the square root of both sides, giving us

The final step is to find the diameter. Since the diameter is twice the radius, we multiply by 2 to find that the diameter is 21 cm.

As discussed earlier, we may not always be given the exact values we need to substitute into the formula for the volume of a cone. One other situation we might have is when we might be given the area or circumference of the circular base, instead of the radius. Recall that both the area and circumference are related to the radius of a circle through the following formulas: where is the area, is the circumference, and is the radius. Thus, in any situation where we are given the area or the circumference instead of the radius, it is always possible to determine the radius.

Let us consider an example of how we might use the above formulas with the volume of a cone, this time phrased as a real-world problem.

Example 4: Finding the Height of a Cone in a Real-World Context

A piece of chocolate is in the form of a right cone; its volume is cm3, and the perimeter of its base is cm. Find its height.

### Answer

In this example, we have been given the volume of a cone and the perimeter of the base and are asked to find the height. Recall that we have the following formula for the volume of a cone: where is the volume, is the radius of the base, and is the height. Although this formula involves the radius and not the perimeter, it is possible for us to find the radius using the fact that the base of a cone is circular. Recall that the perimeter (or circumference) of a circle is given by

Since the perimeter is equal to , we have

By dividing both sides by , we then find that

Now that we have both and , we can substitute these into the formula for the volume to get

We can then rearrange this to find . First, we multiply both sides by 3 to get

We then divide both sides by :

Finally, we divide both sides by to get , which we write in centimetres as follows:

In the previous example, we demonstrated how the circumference of the base can be used together with the volume formula, so let us now consider the case of the area of the base.

As it turns out, if we are given the area and the height, we can even simplify the working further. Since the volume of the cone is , we can substitute to get the following alternate formula for the volume.

### Formula: Volume of a Cone (Base Area)

The volume of a cone is equal to where is the base area and is the height.

We note that this formula is analogous to the alternate formula for the volume of a cylinder of

As expected, this corresponds to our earlier assertion that the volume of a cone is one-third of that of a cylinder.

Let us consider solving an example where we use the area of the base of a cone to find the volume.

Example 5: Finding the Volume of a Cone Where the Area of the Base and the Slant Height Are Given

Find the volume of a right circular cone in terms of if the area of the circular base is cm2 and the slant height is 17 cm.

### Answer

Recall that the volume of a cone is given by where is the radius and is the height. Here, we have been given neither the radius nor the height, but let us recall that since the base of a right circular cone is a circle, we can express the area of the base in terms of the radius by using

Ordinarily, we would just substitute this into the formula for the volume, but in this case, we should be aware that it is necessary for us to find the radius anyway in order to calculate the height. So, let us find the radius by rearranging. We have

Now, we need to find the height. Since the apex of a right circular cone is directly above the center of the base, the height of a cone is directly related to the radius and slant height, as shown below.

Thus, using the Pythagorean theorem, we have

Now, since we have and , we can substitute these values into the equation for the volume of a cone. This gives us

Let us finish by recapping what we have learned in this explainer.

### Key Points

- The volume of a cone is equal to

where is the radius of the base and is

the height. - This formula can be used to find the volume, the radius, or the height of a cone depending on which components we have been given.
- In situations where we are not directly given the height or radius, we can

find them using the slant height by using the Pythagorean theorem as shown

below. - We can also find the radius using the area or circumference of the base. In particular, we can use the alternate formula for the volume of a cone of

where is the base area and is the

height.