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Nathaniel Hammen

Calculus and the volume of a Cone

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First off, because this may look like a homework question, I will tell everyone right now that it's not. Now to the question! I'm taking calculus right now, and we just learned how to calculate the volume of shapes. We used circular cross sections of a cone to come up with the formula for the volume of a cone. I decided to, just for fun, calculate the same volume using triangular cross sections of a cone. Only there was a slight problem... I got a different formula. Pi never even entered the equation anywhere. So, before I show the mathematics (because I left those 3 or 4 lines of math in the car and am too lazy to rewrite even that small amount right now), does anybody know what I did wrong?

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Guest Anonymous Poster
How did you divide a cone into triangular cross sections? You get circular cross sections by slicing the cone one way. By slicing it in a perpendicular direction, you'd actually get cross sections that are part of one lobe of a hyperbola. The only triangle is the slice that is right down the middle of the cone.

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Quote:

Original post by Nathaniel Hammen
Only there was a slight problem... I got a different formula.


actually you should start worrying if you found the *same* formula! :)
Why do you expect to find the same volume? It's not a cone anymore, it's a tetrahedron (sth like a pyramid). Unless I'm misunderstanding sth here...

Post the equation when you get the chance, 'cause I can't help you any more.

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I don't see how you could use a triangular cross section without leaving out some of the volume. The cross section of a single cone is going to be a 2D conic such as a circle, elipse, parabola, or part-hyperbola.

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That's what I get for going ahead. As soon as I saw the first response I had a Homer Simpson moment. I already knew about conic sections, but I was using triangles! I feel extremely stupid now. Hmmm... hyperbolas will be a little more challenging, but I'll still try it.

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