The shape shown in Figure 1 is a pattern for a pendant - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2

To find the radius ( r ) of the circle C that touches the two straight edges and arc AB, we can use a relationship derived from the geometry of the figure:

By trigonometry, the radius is given as:

$\sin(30^\circ) = \frac{r}{6 - r}$

Here, substituting ( \sin(30^\circ) = \frac{1}{2} ), we have:

$\frac{1}{2} = \frac{r}{6 - r}$

From this, we can solve:

$6 - r = 2r$
$6 = 3r \Rightarrow r = 2 \text{ cm}$.

To find the area of the shaded region, we subtract the area of circle C from the area of the sector OAB:

First, calculate the area of circle C:

$\text{Area of Circle C} = \pi r^2 = \pi (2)^2 = 4\pi \text{ cm}^2$

Thus, the area of the shaded region is:

$\text{Area of the Shaded Region} = \text{Area of Sector OAB} - \text{Area of Circle C}$

Substituting the values:

$= 6\pi - 4\pi = 2\pi \text{ cm}^2$.