How to prove formula for volume of a sphere ?

The volume of radius is   × π × radius³  Proved

Explanation:

Given as :

We know that volume of sphere is v = × π × radius³

Or, v =   × π × r³

Let prove the volume of sphere

So, From the figure of sphere

At the height of z , there is shaded disk with radius x

Let Find the area of triangle with side x , z , r

From Pythagorean theorem

x² + z² = r²

Or, x² = r² -  z²

Or, x =

Now, Area of shaded disk = Area = π × x²

Where x is the radius of disk

Or, Area of shaded disk = π × () ²

∴ Area of shaded disk = π × (r² - z²)

Again

If we calculate the area of all horizontal disk, we can get the volume of sphere

So, we simply integrate the area of all disk from - r to + r

i.e volume =

Or, v = -

Or, v = π r² (r + r) -  π

Or, v = π r² (r + r) - π

Or, v = 2πr³ - π

Or, v =  2πr³ ()

Or, v = 2πr³ ×

∴ v =   × π × r³

Hence, The volume of radius is   × π × radius³  Proved . Answer