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Why is the volume of a sphere $\frac{4}{3}\pi r^3$?

We know that the surface area of a sphere is $4\pi r^2$ and the volume is $\frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.

How does one prove these formulae?

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marked as duplicate by Hans Lundmark, Thomas, rschwieb, Ross Millikan, Norbert Oct 21 '12 at 16:56

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

... and see also… – Hans Lundmark Oct 21 '12 at 15:16

For the area, use the equation of a circle of radius $r$, $x^2+y^2=r^2$, to find the area between two curves.

For the volume, view the sphere of radius $r$ as a solid of revolution of the function $y=\sqrt{r^2-x^2}$.

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In this answer I give a geometric derivation for the surface area of a sphere, then integrate that by shells to get the volume of the sphere.

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