A cone is a three-dimensional geometric shape whose circular base narrows smoothly to a point called the apex or vertex of the cone. It can be seen as a set of non-congruent circular disks stacked on one another such that the ratio of the radius of adjacent disks remains constant.
The line segment joining the vertex of the cone to the centre of a circular base is called the height (h) of the cone. The radius of the circular base is called radius (r) of cone. And the length of line segment joining the vertex to any point on the circumference of the circular base is called the slant height (l) of the cone.
A right circular cone is a cone with its vertex above the circular base. When it is not mentioned as a ‘cone’ then usually it is referred to as a right circular cone.
Examples of Cones are: Pencil tip, Funnel, Ice cream cone, Birthday cap, etc.
In this article, you will learn about the volume of cones, its formula and derivation along with some solved examples.
Volume of Cone
In geometry, the volume of any shape is the space occupied by it in a 3-dimensional space. So, the volume of a cone is defined as the space occupied by the cone in a three-dimensional space.
The volume of cone having base radius ‘r’ and height ‘h’ is given by:
Volume of Cone =\[\frac{1}{3}\]πr2h Cubic Units
It can be seen from the above formula that the volume of a cone is one-third of the volume of a cylinder.
Derivation of Volume of Cone
Let us perform an activity to find the volume of cones.
Take a cylindrical container and a conical flask of the same height (h) and same base radius (r).
Add water to the conical flask such that it is filled to the brim.
Now, empty the water filled in a conical flask by pouring it into the cylindrical container. You will notice that the cylindrical container doesn’t fill up fully.
Repeat the above activity once more, you will again notice that the cylindrical container doesn’t fill up fully and have some vacant space.
Now, Repeat the activity for the third time; this time you will notice that the cylindrical container is completely filled.
Thus, we can say that the volume of a cylindrical container is three times the volume of a conical flask. In other words, the volume of conical flask is equal to one-third of the volume of a cylindrical container having the same base radius (r) and height (h).
i.e. volume of cylindrical container = 3 × (volume of conical flask)
or, volume of conical flask = \[\frac{1}{3}\] × (volume of cylindrical container)
=\[\frac{1}{3}\] × πr2h cubic units
Therefore, the volume of cone = \[\frac{1}{3}\] πr2h cubic units
Solved Examples:
Q1. Find the Capacity of a Conical Vessel whose Base Radius is 7 cm and Height is 9 cm.
Solution: Given, radius (r) of conical vessel = 7cm and height (h) of conical vessel = 9 cm.
The capacity of conical vessel = volume of conical vessel
= \[\frac{1}{3}\]πr2h cubic units
=\[\frac{1}{3} \times \frac{{22}}{7} \times {\left( 7 \right)^2} \times 9{\text{ }}c{m^3}\]
= 462 cm3
Therefore, the capacity of a conical vessel is 462 cm3.
Q2. If the Volume of a Right Circular Cone of Height 6 cm is 32 π cm3, find the Diameter of its Base.
Solution: Given, height (h) of cone = 6 cm
Let the base radius of the cone be r cm.
Volume of cone = 32 π cm3
Or, 13 πr2h = 32 π
Or, \[\frac{1}{3} \times \pi \times {\left( r \right)^2} \times \]6 = 32 π
Or, r2 = 16
Or, r = 4 cm
Diameter = 2 r = 2 \[ \times \]4 cm = 8 cm
Therefore, the diameter of the base of the right circular cone is 8 cm.
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