Finding the Volume of a Sphere from its Surface Area

The surface of the sphere is divided into ‘squares’ by the grid lines. The four vertices of each square are joined to the centre of the sphere to make a three dimensional shape that approximates a pyramid. The perpendicular height of each ‘pyramid’ is equal to the radius, r, of the sphere, so the volume of each ‘pyramid’ is approximately ¹/₃ × area of the base × r.

The total surface area of the sphere is  so this is also the total area of all the bases of the 'pyramids'.

Therefore, the total volume of all the pyramids is approximately equal to  .
As the sphere is cut into more and more pieces, the shapes become closer and closer to true pyramids.
In the limit, the volume of the sphere will be exactly  .