# How do I find the volume of a sphere in terms of pi?

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12
Feb 24, 2016

The volume of a sphere of radius $r$ is $\frac{4}{3} \pi {r}^{3}$

#### Explanation:

The surface area of a sphere of radius $r$ is $4 \pi {r}^{2}$.

Imagine dividing a sphere into a large number of slender pyramids with base at the surface and top at the centre of the sphere.

The base of each pyramid will not be quite flat, but the more pyramids you divide the sphere into, the flatter the base of each will be.

Each pyramid has a volume equal to $\frac{1}{3} \text{base" xx "height}$, with the height being equal to $r$, the radius of the sphere.

The sum of the areas of all the bases of the pyramids is equal to the surface area of the sphere (ignoring the slight curvature of the bases).

So the total volume of all the pyramids will be equal to:

$\frac{1}{3} \times 4 \pi {r}^{2} \times r = \frac{4}{3} \pi {r}^{3}$

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seph Share
Oct 17, 2014

The volume $v$ of a sphere in terms of $\pi$ is

$v = \frac{4}{3} \pi {r}^{3}$

why does it need to be multiplied by $\frac{4}{3}$?
the formula was based from calculus. Since the question is under pre-calculus, best accept it as a fact for now.

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