What is the formula for the volume of a sphere of radius #r# ?

1 Answer
Jan 22, 2018

The volume of a sphere is #4/3 pi r^3# where #r# is the radius.

Explanation:

Given that the circumference of a circle of radius #r# is #2pir#, we can deduce that the area of a circle is #pi r^2#. One way to see this is to subdivide the circle into small segements, then rearrange them head-to-tail into a sort of bumpy parallelogram. The more segments the better, but here's an animated illustration with just 8 segments...

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Similarly, if we know that the surface area of a sphere of radius #r# is #4pir^2#, then we can dissect the sphere into small pyramids, with bases tessellating the surface and apices at the centre of the sphere. The height of each pyramid is #r# and if its base area is #A# then its volume is #1/3 xx "base" xx "height" = 1/3 * A * r#. So the total volume of all the pyramids is #1/3 (4pir^2) r = 4/3 pir^3#.