The diagram shows a regular decagon (ten-sided shape with all sides equal and all interior angles equal) - HSC - SSCE Mathematics Advanced - Question 22 - 2020 - Paper 1

To find the angle ( \angle OAB ), we know that the total angle around point O is 360 degrees and there are 10 equal angles in a decagon:

$$\angle OAB = \frac{360}{10} = 36^{\circ}$$

In triangle OAB, we can determine ( \angle AOB ) as:

$$\angle AOB = 180^{\circ} - 36^{\circ} = 144^{\circ}$$

Using the sine rule:

$$\frac{r}{\sin 72^{\circ}} = \frac{8}{\sin 36^{\circ}}$$

Solving for ( r ):

$$r = \frac{8 \cdot \sin 72^{\circ}}{\sin 36^{\circ}} \approx 12.944 ext{ cm}$$

Finally, the area of triangle OAB is given by the formula:

$$\text{Area} = \frac{1}{2} \times OA \times OB \times \sin(\angle AOB)$$

Substituting the values:

$$\text{Area} = \frac{1}{2} \times 12.944 \times 12.944 \times \sin(144^{\circ}) \approx 49.2 \text{ cm}^2$$