15. The shaded area shown in the diagram consists of the interior of a circle of radius 3 together with the area between the circle and two tangents to the circle. The angle between the tangents at the point where they meet is $60^{\circ}$.
What is the shaded area?
A $6 \pi+9 \sqrt{3}$
B $15 \sqrt{3}$
C $9 \pi$
D $9 \pi+4 \sqrt{3}$
E $6 \pi+\frac{9 \sqrt{3}}{4}$
Solution
A
Let $O$ be the centre of the circle and let $P S$ and $P T$ be the tangents to the circle, as shown.

In the triangles $P S O$ and $P T O$ we have $\angle P S O=\angle P T O=90^{\circ}$ because $P S$ and $P T$ are tangents to the circle; $S O=T O$, because they are radii of the same circle; and the side $P O$ is common.

Therefore, the two triangles are congruent (RHS). Hence $\angle S P O=\angle T P O=30^{\circ}$.
It follows that $\frac{O S}{P S}=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$.
Hence $P S=O S \sqrt{3}=3 \sqrt{3}$.
Using the formula area $=\frac{1}{2}$ (base $\times$ height $)$, for the area of a triangle, it follows that the area of the triangle $P S O$ is $\frac{1}{2}(O S \times P S)=\frac{1}{2}(3 \times 3 \sqrt{3})=\frac{1}{2}(9 \sqrt{3})$.
This is also the area of the congruent triangle $P T O$. Therefore the sum of the areas of these two triangles is $9 \sqrt{3}$.
The total shaded area is the sum of the areas of these two triangle plus the area of that part of the circle that lies outside the two triangles. Because $\angle S O T=120^{\circ}$, the part of the circle outside the two triangles makes up two-thirds of the circle and hence its area is given by
$$
\frac{2}{3}\left(\pi \times 3^{2}\right)=6 \pi \text {. }
$$

Hence the shaded area is $6 \pi+9 \sqrt{3}$.
FOR INVESTIGATION
15.1 Explain why $\angle S O T=120^{\circ}$.
15.2 Explain why it follows that the area of the part of the circle outside the two triangles is two-thirds of the total area of the circle.
15.3 The solution above uses the fact that $\tan 30^{\circ}=\frac{1}{\sqrt{3}}$.
Explain why this is correct.
15.4 What is the shaded area in the case where $\angle S P T=120^{\circ}$ ?