25. The cube shown has sides of length 2 units. Holes in the shape of a hemisphere are carved into each face of the cube. The six hemispheres are identical and their centres are at the centres of the faces of the cube. The holes are just large enough to touch the hole on each neighbouring face. What is the diameter of each hole?
A 1
B $\sqrt{2}$
C $2-\sqrt{2}$
D $3-\sqrt{2}$
E $3-\sqrt{3}$
SolUTION
B

Let $P$ and $Q$ be the centres of two adjacent hemispheres. The faces on which these hemispheres are carved meet at an edge. Let $M$ be the midpoint of that edge. Then $M P=M Q=1$. Also $M P Q$ is a right-angled triangle since the two faces are perpendicular. By Pythagoras, $M P^{2}+M Q^{2}=P Q^{2}$, so $P Q^{2}=1+1=2$. Hence $P Q=\sqrt{2}$, and $P Q$ is equal to the sum of two radii, so is the same as the diameter of the hemispheres.