I want to find the volume of the cut-cone in the picture with respect to $R, r$ and $h$. Can you check my work

The volume of the cut-cone is is the volume of the full cone minus the volume of the small cone which gone so $V = \dfrac{\pi}{3} R^2 x - \dfrac{\pi}{3} r^2 (x-h) $

But from the similarity of the triangles we get $\dfrac{x}{R} = \dfrac{x-h}{r} \rightarrow x = \dfrac{hR}{R-r}$

$V = \dfrac{\pi}{3} \left(x(R^2 - r^2) + rh \right)=\dfrac{\pi}{3}\left(\dfrac{Rh(R^2-r^2)}{R-r} +r^2h \right)= \dfrac{\pi h}{3}(R(R+r)+ r^2) $

enter image description here

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    $\begingroup$ Seems pretty accurate to me , what seems to be the question here ? $\endgroup$ – Sujith Sizon Oct 21 '15 at 4:40
  • $\begingroup$ just check my work $\endgroup$ – Ameryr Oct 21 '15 at 4:51
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    $\begingroup$ If you google a bit you can find some sites with the formulas for the volume. For example, MathWorld. $\endgroup$ – Martin Sleziak Oct 21 '15 at 5:49
  • $\begingroup$ ah thanks I like to do it by myself so I will never forget it. $\endgroup$ – Ameryr Oct 21 '15 at 10:35

Man what's the question the cone is more like a frustum and volume of small cone =volume of original cone - volume of frustum $= 1/3πR^2x-1/3π(R^2+r^2+rR)(x-h)$. Here $h$ is height of frustum.

  • $\begingroup$ yea it is the frustrum $\endgroup$ – Ameryr Oct 21 '15 at 4:51
  • $\begingroup$ But doesn't cut-cone mean the frustum ? $\endgroup$ – Sujith Sizon Oct 21 '15 at 5:33

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