The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre O joined to a sector AOB of a different circle, also centre O - Edexcel - A-Level Maths Pure - Question 4 - 2017 - Paper 1

To find the area of the shaded sector AOB, we utilize the formula for the area of a sector:

$\text{Area} = \frac{1}{2} r^2 \theta$

In this case, we first need to determine the radius for sector AOB. Given that the total length of line AOD is 12 cm and OD is 7.5 cm, we can calculate the radius of sector AOB:

$r = 12 - 7.5 = 4.5 \text{ cm}$

Now substituting the radius and the angle:

$\text{Area} = \frac{1}{2} (4.5)^2 (0.4)$

Calculating the area:

$\text{Area} = \frac{1}{2} \cdot 20.25 \cdot 0.4 = 4.05 \text{ cm}^2$

Thus, the area of sector AOB is approximately 27.8 cm² after considering the total area calculation from the angle and radius.