CBSE Class 10 Answered

--------------------- +P --------------------
       |                  /|\                  |
       |                 / | \                 |
       |                /  |  \ S-s            |H-h
       |               /   |   \               |
       |              /    |    \              |
       |             /*****|**r**\             |
       |            **    C+-----*+D--------------
      H|         S /  ***********  \           |
       |          /        |        \          |
       |         /         |         \         |
       |        /          |          \ s      |
       |       /           |           \       |h
       |      /            |            \      |
       |     /             |             \     |
       |    /    **********|**********    \    |
       |   /*****          |       R  *****\   |
     -----*               A+----------------+B----
           ******                     ******
                 *********************

We know R, r, and h, but not H, the total height of the cone from 
which the frustum was cut. If we can find it, then the volume of the 
frustum will be the volume of the whole cone, pi R^2 H/3, minus the 
volume of the cone we cut off the top, pi r^2 (H-h)/3.

The triangles PAB and PCD are similar, so we can write the equation

     AB   CD        R    r
     -- = --   or   - = ---
     PA   PC        H   H-h

Cross-multiplying [that is, multiplying both sides by H(H-h)], we get

      R(H-h) = rH

We can distribute the left side and collect H terms, then divide:

     RH - Rh = rH

     RH - rH = Rh

      (R-r)H = Rh

               Rh
           H = ---
               R-r

Now let's write the volume formula and substitute this formula for H:

         pi         pi
     V = -- R^2 H - -- r^2 (H-h)
          3          3

         pi
       = -- (R^2 H - r^2 H + r^2 h)
          3

         pi
       = -- [(R^2 - r^2) H + r^2 h]
          3

         pi              Rh
       = -- [(R^2 - r^2) --- + r^2 h]
          3              R-r

         pi               R
       = -- [(R^2 - r^2) --- + r^2] h
          3              R-r

We can write R^2 - r^2 as (R - r)(R + r) and cancel:

         pi
       = --- [(R + r) R + r^2] h
          3

         pi
       = --- [R^2 + Rr + r^2] h
          3

That's the formula.