Here you will learn what is the formula for volume of cone and examples based on it.
Let’s begin –
What is Cone ?
A cone is a solid which has a circle at its base and a slanting lateral surface that converges at the apex. Its dimensions are defined by the radius of the base (r), the height (h) and the slant height (l).
Also Read : Surface Area of Cone – Formula & Examples
Formula for Volume of Cone
Volume of Cone = \(1\over 3\) \(\pi r^2 h\)
where r is the base radius and h is the height of the cone.
Example : The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone.
Solution : Here h = 21 and \(l\) = 28
We know that, \(l^2\) = \(r^2\) + \(h^2\).
\(\implies\) \(r^2\) = \(l^2\) – \(h^2\)
\(\implies\) r = \(\sqrt{ 28^2 – 21^2}\) = \(7\sqrt{7}\)
So, Volume of the Cone = \(1\over 3\) \(\pi r^2 h\)
= \(1\over 3\) \(\pi \times 343 \times 21\)
= 7546 \(cm^3\)
Example : In a marriage ceremony of her daughter Poonam, Ashok has to make arrangements for the accommodation of 150 persons. For this purpose, he plans to build a conical tent in such a way that each person has 4 sq. meters of the space on ground and 20 cubic meters of air to breath. What should be the height of the conical tent ?
Solution : Let the height of the conical tent = h metre.
Radius of the base of the cone = r meter.
The tent has to accommodate 150 persons.
The space required by each person on the ground = 4 \(m^2\)
And the amount of air = 20 \(m^3\)
\(\therefore\) Area of the base = 150 \(\times\) 4 = 600 \(m^2\)
\(\implies\) \(\pi r^2\) = 600 \(\implies\) r = 13.817 m
Volume of the air required for 150 persons = 150 \(\times\) 20 = 3000 \(m^3\)
\(\implies\) \(1\over 3\) \(\pi r^2 h\) = 3000 \(m^3\)
\(\implies\) h = \(3000 \times 7 \times 3\over 22 \times (13.817)^2\) = 15 m
Hence the height of the conical tent is 15 m.
