Volume of A Cone

A cone is a three-dimensional geometric shape that tapers from a flat base to a point called apex or vertex, and here we will learn how to find out the volume of the cone. A cone is formed by a set of line segments, half-lines or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex.

Cone is a three-dimensional structure having a circular base. A cone can be seen as a set of non-congruent circular disks that are stacked on one another such that ratio of the radius of adjacent disks remains constant. You can think of a cone as a triangle which is being rotated about one of its vertices. Now, think of a scenario where we need to calculate the amount of water that can be accommodated in a conical flask. In other words, we mean to calculate the capacity of this flask. The capacity of a conical flask is basically equal to the volume of the cone involved. Thus, Volume of a three-dimensional shape, in general, is equal to the amount of space occupied by that shape. Let us perform an activity to calculate the volume of a cone,

Take a cylindrical container and a conical flask of the same height and same base radius. Add water to the conical flask such that it is filled to the brim. Start adding this water to the cylindrical container you took. You will notice it doesn’t fill up the container fully. Try repeating this experiment for once more, you will still observe some vacant space in the container. Repeat this experiment once again; you will notice this time the cylindrical container is completely filled. Thus, the volume of a cone can be said to be equal to one-third of the volume of a cylinder having the same base radius and height.

In general, a cone is a pyramid with a circular cross section. A right cone is a cone with its vertex above the surface. When it is not mentioned a ‘cone’ is referred as a ‘right cone.’

You can easily find out the volume of the cone if you have the measurements of its height and radius and put it into a formula.

volume of cone

Volume of a cone = 1/3 πr2h

How is the volume of cone and volume of a cylinder related?

The volume of a cone is 1/3 πr2h where r is the radius of the cone at the wider end of the cone. Look how similar the formula of a cone is the formula of a cylinder. As you can see only 1/3 is extra here. What’s that for? Well, one end is a circle, and one is a point. So that means 1/3 of the volume of a cylinder is a cone. In other words, you need three cones to make one cylinder.

volume of a cone

Let’s Work Out:

Example: Calculate the volume if r= 2 cm and h= 5 cm.

Solution:

Given:

r = 2

h= 5

Using the formula of Volume of Cone

= \(\frac{1}{3}\pi r^{2}h\)

= \(\frac{1}{3} \times 3.14 \times 2^{2} \times 5 \)

= \(\frac{1}{3} \times 3.14 \times 4 \times 5 \)

= \(\frac{1}{3} \times 3.14 \times 20 \)

= \(20.93 \;\; cm^{3}\)

Example: Calculate the volume of a cone having base equals to 7 cm and slant height equals to 15 cm. (Take \(\pi\) = \(3.14\))

Solution: Volume of a cone = \(\frac{1}{3}\pi r^2 h\) ……….(i)

Using Pythagoras Theorem, we have

\(l^{2} = h^{2} + \left ( \frac{b}{2} \right )^{2}\)

\(\Rightarrow h^{2} = l^{2} + \left ( \frac{b}{2} \right )^{2}\)

\(\Rightarrow h= \sqrt{l^{2} + \left ( \frac{b}{2} \right )^{2}}\)

\(\Rightarrow h= \sqrt{15^{2} + \left ( \frac{7}{2} \right )^{2}}\)

\(\Rightarrow h= 14.58\)

Substituting the value of h in equation (i), we have

\(V = \frac{1}{3} \times 3.14 \times (3.14)^2 \times 15\)

V = 154.8 \(3.14 \;\; cm^{3}\)<


Practise This Question

Which of the following shows  12 and 112 on the number line?