A solid sphere is cut into two hemispheres. From one, a square pyramid and from the other a cone. each of maximum possible size are carved out. What is the ratio of their volumes?

Question

Philoid · Accepted Answer

1. Consider the red hemisphere in fig (a) below.

• A dotted ellipse and a dashed curve are drawn inside it

• The dashed ellipse represents the base of the hemisphere

• For maximum possible volume, this base is taken as the base of the cone in fig. (b)

• We can see that, the cone fits perfectly in the hemisphere.

• This is shown more clearly in (c)

2. From fig.c, we have:

• Height of the cone, = r

• Radius of the cone, = r

• So Volume,

3. Consider the red hemisphere in fig (a) below. It is the same hemisphere of radius r, that we saw for the cone above

• A square is drawn in the base of the hemisphere

• This square is the base of the pyramid

4. For maximum possible volume, the diagonal of the square must be equal to the diameter of the circle

• So in fig. c, we can write:

OP = OQ = half of diameter = radius = r

5. OPQ is a right triangle. We can apply Pythagoras theorem

• Then base edge = PQ = = = = r

6. So volume of the pyramid, =

7. Now we can take the ratio:

• Thus we get:

: = 2 : π