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Given a right triangle $ABC$ rotated $2\pi$, one full circle, around line $BC$. Find area of the formed cone.

Everyone knows that the formula for a cone includes a $\frac13$ in it due to some integral calculus.

Why can the area of a cone not be visualized as a triangle rotated about an axis? Intuitively this makes sense. The area of a triangle is $\frac12bh$. The distance it is rotated is $2\pi$. By that this should give us a formula $bh\pi$. This seems to work for a cone with $r = 3$ and height $4$. (Aka a $3,4,5$ Right Triangle Rotated).

Is there a non-calc based approach for understanding why this does not work?

David
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Ean Dudley
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  • Yes rotation of triangle will give you a cone and it does work - you can show that using calculus. – Math Lover Mar 08 '21 at 05:13
  • You appear to be mixing together concepts related to volume with concepts related to surface area. The factor of $\frac13$ appears in the formula for the cone's volume, not its surface area. And to find the (curved) surface area of the cone from the rotated triangle, you need to consider the surface "swept out" by a side of the triangle. Using the triangle's area will give you the cone's volume. OTOH, it's easier to get the curved surface area of the cone by showing that it can be "unwrapped" to a sector of a circle with radius equal to the slant height of the cone. – PM 2Ring Mar 08 '21 at 06:11

2 Answers2

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The area is the area of the base ($\pi r^2$ where $r = \overline{AB}$) and the area of the curved portion is $\pi R^2 \theta$ where $R = \overline{AC}$ and $\theta$ is the angle subtended by the surface if flattened onto a plane. The length of the perimeter is $\pi r^2$ and that is the proportion of the full $2 \pi$ central angle by $\frac{2 \pi r}{2 \pi R}$.

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David G. Stork
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First note that the triangle is not rotated through "a distance of $2\pi$" but through an angle of $2\pi$, which is not the same thing.

Also, you seem to be assuming that the volume will be the area multiplied by the angle of rotation, which is not true.

A similar result which is true is Pappus' theorem, which states that the volume swept out by a plane figure when rotated is equal to its area, multiplied by the distance travelled by its centroid.

David
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