Lesson EASY PROOF of volume of a sphere

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This proof was known to ancient greeks and does not involve calculus or integration.

<B>Take a hemisphere</B>.

<B>Surround it by a cylinder</B> of the same radius as the hemisphere, and the same height as the height of the hemisphere. We assume you know the volume of this cylinder: volume is area of the base multiplied by height. Note that the height is the same as the radius of the base:
{{{Volume(cylinder) = pi*r^2*r = pi*r^3}}}

<B>Take an upside down right circular cone in the cylinder</B>. The 'base' of the cone will be at the top of the cylinder, and the point at the bottom will be at the center of the hemisphere. The volume of a cone is {{{V(cone) = expr(1/3)*Area( base )*height = expr(1/3)*pi*r^2*r}}}.

{{{
drawing( 400, 200, 0, 6, 0, 3,
  red( circle( 3, 0, 2 ), locate( 2.0, 1.9, Sphere ) ),
  blue( rectangle( 1, 0, 5, 2 ), locate( 4, 1.95, cylinder ) ),
  green( line( 1, 2, 3, 0 ), line( 5, 2, 3, 0 ), locate( 1, 1.5, Cone ) )
)
}}}

<H4>Proposition</H4>
 
On any horizontal slice of this configuration, the area of the cross section of the hemisphere
is the DIFFERENCE between the area of the cross section of the cylinder MINUS the area of the cross section of the inverted cone.

{{{Area(slice_of_hemisphere) = Area( slice_of_cylinder ) - Area( Cone )}}}

<H4>Proof of the proposition</H4>

Suppose that we made out "slice" at certain height h.

Radius of the slice of hemisphere would be {{{Radius(hemisphere_slice) = sqrt( r^2-h^2 )}}}. The area will be, therefore, {{{Area(hemisphere_slice) = pi*(radius( hemisphere_slice ))^2 = pi*(r^2-h^2)}}}.

Area of the slice of the cylinder would be {{{pi*r^2}}}, since the cylinder has radius r regardless of where we make the slice.

Radius of the cone would be h, so the area of the slice would be {{{Area( cone_slice ) = pi*h^2}}}. 
Subtracting the area of the cone slice, from area of the cylinder slice, we get

{{{Area( cylinder_slice ) - Area( cone_slice ) = pi*r^2 - pi*h^2 = pi*(r^2-h^2)}}}

Since {{{pi*(r^2-h^2)}}} is the same as the formula for the area of the hemisphere slice, the proposition is proven.

<H4>Principle of volume</H4>

If two solids have cross sections of equal area for all horizontal slices, then the have the same volume.

<H4>Final result</H4>

Therefore, {{{
volume_of_hemisphere 
= volume_of_cylinder - volume_of_cone
= pi*R^3 - expr(1/3)*pi*R^3
= expr(2/3)*pi*R^3
}}}

The volume of the sphere is twice that, {{{ expr(4/3)*pi*R^3}}}.