Volume of Cone

When you look at a volcanic mountain, what does it look like? Perhaps the ice cream cones are more attractive and you may be wondering how much ice cream they can contain.

In this article, you will understand what a cone and conical objects are, how to determine their volume and examples of application of this.

What is a cone?

Some examples of objects with conical shapes are the traffic cone, birthday cone caps, ice cream cones, carrots and so on.

An image of traffic road cones - StudySmarter Original

An image of an ice cream cone - StudySmarter Original

Types of cones

There are two basic types of cones:

• The right circular cones which have their apex perpendicular to or right above the center of their base.

An illustration of a right circular cone - StudySmarter Original

• The oblique circular cones - do not have their apex above the center of the circular base and as such are not perpendicular to the center.

An illustration of an oblique circular cone - StudySmarter Original

Derivation of the Volume of Cones from the Volume of Cylinders

The volume or capacity of a cone is one-third of its corresponding cylinder. This means that when a cone and a cylinder have the same base dimension and height, the volume of the cone is one-third of the volume of the cylinder. Therefore, a cone can be obtained by dividing a cylinder into three parts. In order to visualize this, we will do a little experiment.

Experiment on deriving cones from cylinders

First, let's understand the following: The volume of a solid can also be understood as its capacity. The capacity of a solid is how much of a liquid – we usually think of water – it can contain.

Now to the experiment. It can be done through the following steps.

Step 1. Get an empty regular cylinder with a known height and radius of the circular base.

Step 2. Get an empty regular cone with its height and radius the same as that of the cylinder.

Step 3. Fill the cone with water to the brim.

Step 4. Pour all the water in the cone into the empty cylinder and observe the level of water in the cylinder.

Note that the cylinder is yet to be filled up.

Step 5. Repeat steps 3 and 4 for the second time on the same cylinder which already has water in it. Notice that the water level rise but the cylinder is yet to be filled to the brim.

Step 6. Repeat steps 3 and 4 for the third time on the same cylinder. Notice that the cylinder is now filled to the brim.

This experiment explains that you would need 3 cones to make a cylinder.

Since we know that a cone is one-third of its corresponding cylinder, it is easier to calculate the volume of a cone.

We recall that the volume of a cylinder is the product of its circular base area and its height and is given by,

$Volum{e}_{Cylinder}=\pi r^2\times h$.

Thus, the volume of a cone is one-third the product of its circular base area and height.

$Volum{e}_{Cone}=\frac{Volum{e}_{Cylinder}}{3}=\frac{\pi r^2\times h}{3}$

Note that the volume of an oblique cone is calculated with the same formula as that of a regular right cone. This is because of Cavalieri's principle which explains that the volume of a solid shape equals the volume of its corresponding oblique object.

How to calculate the volume of a cone without a known height?

Sometimes we may be asked to find the volume of cones without a known height. In this case, we need to look out for either a known slant height or a known angle. With a slant height given, you can use Pythagoras' theorem to calculate it. Therefore, let l be the slant height,

Illustration of a cone with an angle subtended at the apex - StudySmarter Original

When an angle is given, SOHCAHTOA is applied to derive the value of the height. If the angle at the apex is given, then half of that angle is used to find the height. Thus;

where;

θ is the angle at the apex,

l is the slant height of the cone,

h is the height of the cone,

r is the base radius of the cone.

Volume of the Frustum of a cone

An illustration on how a frustum is formed from a cone - StudySmarter Original

To calculate the volume of a frustum, proportion is used. A frustum has two radii - a radius for the bigger circular surface and another radius for the smaller circular surface.

An image showing the derivation of volume of a frustum - StudySmarter Original

Let R be the radius of the bigger circular surface, and r be the radius of the smaller circular surface. Let h_f be the height of the frustum, and H be the height of the cone when complete. And finally, let h_c be the height of the small cone which was cut off to form the frustum.

Therefore, while using proportion we have,

$\frac{R}{r}=\frac{H}{h_c}$

The volume of the frustum is equal to the difference between the volume of the big cone and the volume of the small cut cone, and hence we have

But,

$Volum{e}_{bigcone}=\frac{1}{3}\pi R^{2}H,Volum{e}_{smallcone}=\frac{1}{3}\pi r^2{h}_c$

Thus, the volume of the frustum is

$Volum{e}_{frustum}=\frac{1}{3}\pi R^{2}H-\frac{1}{3}\pi r^2{h}_c$