# How do I determine the equation to find the volume of a right circular cone

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The question is slightly vague as it does not give what exactly is required.

The volume of a right circular cone is given by `(1/3)*pi*r^2*h`

To derive this, the cone has to be broken into infinitesimally thin circular strips of width `dx` . Moving up from the base, the radius of the strips decreases and is equal to `(h - x)*(r/h)` . The volume of each of them is `pi*((h - x)*(r/h))^2 dx` with x varying from 0 to h.

Adding the volume of all the strips gives the volume of the cone. This is given by the integral `int_(0)^h pi*((h - x)*(r/h))^2 dx`

=> `pi*(r/h)^2*int_(0)^h (h - x)^2 dx`

=> `pi*(r/h)^2*int_(0)^h h^2 - 2hx + x^2 dx`

=> `pi*(r/h)^2*(h^2*x - 2*h*x^2/2 + x^3/3)_(0)^h`

=> `pi*(r/h)^2*(h^3 - h^3 + h^3/3)`

=> `pi*h^3*(r^2/h^2)*(1/3)`

=> `(1/3)*pi*r^2*h`

The volume of a right circular cone with height h and base radius r is `(1/3)*pi*r^2*h`