6. Three points, $P, Q$ and $R$ are placed on the circumference of a circle with centre $O$. The arc lengths $P Q, Q R$ and $R P$ are in the ratio $1: 2: 3$.
In what ratio are the areas of the sectors $P O Q, Q O R$ and $R O P$ ?
A $1: 1: 1$
B $1: 2: 3$
C $1: \pi: \pi^{2}$
D $1: 4: 9$
E $1: 8: 27$
Solution
B

The key facts that we need to use here are
(a) the ratio of the length of an arc to the length of the circumference is the same as the ratio of the angle that the arc subtends at the centre of the circle to the angle in a complete revolution, and
(b) the ratio of the area of a sector to the area of the circle is the same as the ratio of the angle in the sector to the angle in a complete revolution.

It follows from (a) that the ratios of the arc lengths of $P Q, Q R$ and $R P$ are the same as the ratios of the angles that the arcs subtend at the centre of the circle.

Therefore these angles are in the ratio $1: 2: 3$.
Similarly, it follows from (b) that the ratios of the areas of the sectors $P O Q, Q O R$ and $R O P$ are the same as the ratios of the angles in the sectors.
Therefore the areas of the sectors $P O Q, Q O R$ and $R O P$ are in the ratio $1: 2: 3$.
Commentary
From the basic facts (a) and (b) we can deduce that
(c) The ratio of the length of an arc of a circle to the circumference of a circle is equal to the ratio of the area the arc subtends at the centre of the circle to the area of the circle.
An alternative method would have been to base the solution on this fact.