How to prove volume of sphere is 4/3 × π × r³

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

Or, v =  × π × r³

Below is the proof:-

So, From the figure of sphere below

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

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

Using Pythagoras’ 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  is proved

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