# How to prove formula for volume of cone ?

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As seen in the below image, the cone is divided into 2 triangles which can be proved similar by AA test.

l is the slant height of the cone, r is the radius

y is the radius of the disc on the smaller triangle with x as height

Using similarity

r            h                          rx

__ =  ______  =>        y =  ________

y           x                            h

By integration its easy to see that volume of cone is summation of smaller discs with area

$\prod y^{2}dx$ integrated over the interval 0 to h

Volume of cone = $\int_{0}^{h}\prod y^{2}dx$

=> $\int_{0}^{h}\prod (\frac{rx}{h})^{2}dx$

=> $\prod \frac{r^{2}}{h^{2}}\int_{0}^{h}x^{2}dx$

=> $\prod \frac{r^{2}}{h^{2}}[\frac{x^{3}}{3}]_{0}^{h}$

=>$\prod \frac{r^{2}}{h^{2}}[\frac{h^{3}}{3}]$

=> $\frac{1}{3}\prod r^{2}h$

Hence proved, volume of cone is = $\frac{1}{3}\prod r^{2}h$

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