Question

How will you find the volume of a cone and a dome?

Answer 1

Hint:
Cone is the three – dimensional geometric shape which has the circular base that tapers from the flat base to the point. Volume of cone is the capacity of the cone whereas the shape of the dome is similar to the hemisphere.

Complete step by step solution:
The volume of the cone can be defined as the capacity of the cone or the space of the cone. The cone is the three – dimensional geometric shape having a circular base that tapers from a flat base to a point which is known as vertex or apex. The cone is formed by the set of line segments, lines connecting a common point or half lines, the apex with all the points which are on the base which is in the plane and does not contain the apex.
So, the formula for the volume of cone is –
$V = \dfrac{1}{3}\pi {r^2}h$
where, $r$ is the base radius of the cone, and
$h$ is the height of cone
The set of points in the three – dimension and all the points lying on the surface is equidistant from the center then, the shape is known as sphere. When we cut the sphere from its center into two equal parts then, those parts are known as hemispheres. The volume of the dome will be equal to the volume of the hemisphere because their shape and size are equal to each other. The volume of the hemisphere was derived by Archimedes. The formula for volume of sphere is –
$V = \dfrac{4}{3}\pi {r^3}$
where, $r$ is the radius of hemisphere
For hemisphere we will divide the volume of sphere by 2 –
$
   \Rightarrow V = \dfrac{4}{3}\pi {r^3} \times \dfrac{1}{2} \\
   \Rightarrow V = \dfrac{2}{3}\pi {r^3} \\
 $
So, the volume of dome will be –
$V = \dfrac{2}{3}\pi {r^3}$
which is the same as the volume of the hemisphere.

Note:
The volume of the cone is one – third of the volume of the cylinder therefore, the volume of the cone will be equal to the one – third of the volume of the cylinder.
The formula for the volume of the right circular cone, oblique cone and the regular cone is the same.