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

### Definition: Cones

Cones are three-dimensional geometric shapes, or solid objects, that have a (generally) circular *base*
and a curved side that ends in a single vertex or apex.

A right cone is a cone whose apex lies above the centroid of the base. (When the base is circular, the apex lies above the circleβs center.)

The **height** of a cone is the distance from the apex to the base.

The *slant height* of a cone is the distance from the apex to any point lying on the circumference of the base.

Now that we have learned what a cone is, letβs look at its volume.

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 would observe that the level of water is exactly at one-third of the height of the cylinder.

This is a general rule for any cone.

### The Volume of a Cone

The volume of a cone is one-third of the volume of a cylinder of the same base and same height:

Remember that the area of a circle of radius is .

Letβs look at some examples.

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

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

### Answer

We know that the volume of a cone is one-third of the volume of a cylinder of the same base and height; that is,

To work out the volume of this cone, we need to find the area of the base, that is, the circle of radius 20 cm. The area is given by . By plugging in the value of , we find

The volume of the cone is

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

Find the volume of the cone. Give your answer in cubic millimeters to two decimal places.

### Answer

The volume of a cone is given by , where is the area of its circular base, and is the height of the cone. We observe that the cone shown in the diagram does not sit on its base. With this in mind, we identify that the height of the cone, the distance between its apex and base, is 63 mm, while the diameter of its base is 58 mm. The area of the circular base is which, knowing that the radius is half the diameter, that is, 29 mm, gives

Susbtituting this and the value of the height into our equation giving the volume of a cone, we find that

Calculating this with a calculator and rounding to two decimal places, we find

### Example 3: 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 the cone, we need to find the area of its circular base. However, we do not have the radius, but we have the height and the slant height of the cone.

Realizing that these two lines and a radius of the circular base form a right triangle (note that we know that the apex is above the center of the base because we are told it is a right cone), we can apply the Pythagorean theorem, with the radius of the circular base:

Taking away 2,304 from each side,

Taking the square root of both sides,

We can now work out the volume of the cone:

### Example 4: Finding the Volume of a Cone given Its Radius and Height

Work out the volume of a cone with a radius of 3 and a height of 14. Give your solution to two decimal places.

### Answer

The volume of a cone is given by , where is the area of its circular base and is the height of the cone. The area of the circular base is , where is the radius of the circular base.

Hence, we have

We are told in the question that the radius is 3 and the height is 14. Plugging in these values, we find that

Using our calculator and rounding our answer to two decimal places, we find that

In this question, no length unit was specified, so it is implicit that all lengths are given in the same length unit, and our result is measured in this unit cubed.

### Example 5: Finding the Diameter of a Cone given Its Volume and Height

The volume of a cone is cubic inches, and its height is 12 inches. Find its diameter.

### Answer

We have here the volume and the height of a cone, and we want to find the coneβs diameter. To do this, we need to write the relationship between the volume, height, and radius of a cone. This will allow us to find the radius of the cone. It is half the diameter, so we will need to double the radius to find the diameter.

We have

Plugging the value of the volume and the height of the cone into the equation, we find

Multiplying by 12 (since the multiplication is commutative),

Dividing both sides by ,

Taking the square root of each side,

The radius of the cone is 10.5 inches, so its diameter is 21 inches.

### Key Points

- Cones are three-dimensional geometric shapes, or solid objects, that have a (generally) circular base and a curved side that ends in a single vertex or apex.
- A right cone is a cone whose apex lies above the centroid of the base. (When the base is circular, the apex lies above the circleβs center.)
- The
**height**of a cone is the distance from the apex to the base. The*slant height*of a cone is the distance from the apex to any point lying on the circumference of the base. - The volume of a cone is one-third of the volume of the cylinder of the same base and same height.