# Thread: volume of cone

1. ## volume of cone

Use integration to write aformula that gives the volume of cone , the radius of whose base is r and whose height is h

2. Originally Posted by BAHADEEN
Use integration to write aformula that gives the volume of cone , the radius of whose base is r and whose height is h
File:VOLUME OF CONE BY INTEGRATION.pdf - Wikimedia Commons

3. Originally Posted by BAHADEEN
Use integration to write aformula that gives the volume of cone , the radius of whose base is r and whose height is h
Alternatively,

$R=radius,\;H=height,\;r=varying\;cross-sectional\;radius,$

$h=height\;of\;internal\;cone\;of\;radius\;r$

$\displaystyle\frac{R}{H}=\frac{r}{h}=constant\Righ tarrow\ R=Hk,\;\;r=hk$

The constant is $tan\theta$ but this form is unnecessary.

$\displaystyle\int_{0}^H{{\pi}r^2}dh={\pi}k^2\int_{ 0}^H{h^2}dh$

$\displaystyle\ ={\pi}k^2\frac{H^3}{3}=\frac{{\pi}R^2}{H^2}\;\frac {H^3}{3}=\frac{{\pi}R^2H}{3}$