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

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Lyla Share
Feb 24, 2016

the formula for a sphere is 3/4pi r cubed
So if the radius was 3 then u would multiply 3 times 3 times 3 and u get 27
Then u multiply 27 by 4 which in this case is 108 then divide 108 by 3 and u get 36
So in terms of pi, the answer would be 36*pi

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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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Apr 5, 2016

Relate the volume of a sphere to the volume of a cylinder.

#### Explanation:

You could compare the volume of a sphere to the volume of a cylinder. If you have a sphere with a radius of 2 and a cylinder with a radius of 2 and a height of 4, then the cylinder will fit perfectly around the sphere and have the same height. If the sphere were a clay ball, then you could mash it into the cylinder to fill in the space. The height of the sphere, is now 2/3 that of the cylinder. Therefore, the volume of the sphere is $\frac{2}{3} \pi {r}^{2} h$. Try it with numbers.
Example:
Sphere has a radius of 2. Cylinder has a radius of 2 and a height of 4.
Sphere: $\frac{4}{3} \pi {r}^{3}$ = $\frac{4}{3} \pi \left({2}^{3}\right)$ = $\frac{4}{3} \pi \left(8\right)$ =$\frac{32}{3} \pi$

Using the derived formula: $\frac{2}{3} \pi {r}^{2} h$ = $\frac{2}{3} \pi \left({2}^{2}\right) \left(4\right)$ = $\frac{2}{3} \pi \left(4\right) \left(4\right)$ = $\frac{32}{3} \pi$
The "h" in the derived formula is actually the diameter of the sphere. And can be rewritten as 2r which would make the derived formula then become $\frac{2}{3} \pi \left({r}^{2}\right) \left(2 r\right)$ which actually equals $\frac{4}{3} \pi {r}^{3}$.

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