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It is probably dublicated but couldnt find. Let radius=r and height=h, find the volume of oblique cone with use of integral

if it was right cone I'd use $y=h-\frac{xh}{r}$ so $\pi\int_0^h(\frac{r(h-y)}{h})^2dy$.

is it same with oblique?

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3 Answers 3

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The transformation which maps a right cone onto a oblique cone is described by the shear matrix which looks like so (k parameter determines the obliques):

$$ \left(\matrix{1\;0\;k\\0\;1\;0\\0\;0\;1}\right)$$

It has the determinant equal to $1$ therefore the formula for the cone's volume doesn't change. It remains

$$ V= \frac{hr^2\pi}{3} $$

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  • $\begingroup$ For shearing from right circular cone to oblique circular cone, the shear matrix should be $$ \left(\matrix{1\;0\;k\\0\;1\;h\\0\;0\;1}\right)$$ $\endgroup$
    – Black Mild
    Oct 30, 2021 at 3:10
  • $\begingroup$ We can pick the coordinate axis so that the shear is axis aligned $\endgroup$
    – Michal
    Nov 3, 2021 at 4:35
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I want to integrate the cone from tip to base. What I would do is first find the limits of integration, 0 to h. Then I find the area of each individual circle. $\pi*\text{radius}^2$

I substitute how the radius changes for $\text{radius}$. As y changes from 0 to h, the radius is $\frac{ry}{h}$.

So... the integral is $$ \int_0^h{\pi\left(\frac{ry}{h}\right)^2 \;dy }$$

Its the same; you just integrated from base to tip while I did tip to base.

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  • $\begingroup$ Welcome to SE.maths! Here's a MathJax Tutorial to help you write with the correct syntax. $\endgroup$
    – user88595
    Jun 2, 2014 at 22:53
  • $\begingroup$ is it enough to say volume of right and oblique cone are equal, how do we prove it simply $\endgroup$
    – lyme
    Jun 3, 2014 at 13:55
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It will be same as the volume of a right cone. Since cones formed on equal base area and between same parallel planes have equal volumes. Its similar to the theorem of triangles formed on equal bases and between same parallel lines.

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