The circle below has center O, and its radius is 9m. Given that =m∠AOB40°, find the area of the shaded region and the length of the arc ADB.

1. Break Down the Problem

To find the area of the shaded region and the length of the arc ADB, we will:

1. Calculate the area of sector AOB.
2. Calculate the length of arc ADB.

2. Relevant Concepts

• Area of a sector: $A = \frac{\theta}{360^\circ} \times \pi r^2$
• Length of an arc: $L = \frac{\theta}{360^\circ} \times 2\pi r$

Where:

• $\theta$ is the angle in degrees,
• $r$ is the radius of the circle.

3. Analysis and Detail

Step 1: Calculate the area of sector AOB

Using the formula for the area of a sector:

$A = \frac{40^\circ}{360^\circ} \times \pi \times (9m)^2$

Calculating this:

$A = \frac{40}{360} \times \pi \times 81$

$A = \frac{1}{9} \times \pi \times 81$

$A = 9\pi \, \text{m}^2$

Step 2: Calculate the length of arc ADB

Using the formula for the length of an arc:

$L = \frac{40^\circ}{360^\circ} \times 2\pi \times 9m$

Calculating this:

$L = \frac{40}{360} \times 2\pi \times 9$

$L = \frac{1}{9} \times 2\pi \times 9$

$L = 2\pi \, \text{m}$

4. Verify and Summarize

We calculated:

• The area of the shaded region (sector AOB) is $9\pi \, \text{m}^2$.
• The length of the arc ADB is $2\pi \, \text{m}$.

Final Answer

• Area of the shaded region: $9\pi \, \text{m}^2$
• Length of the arc ADB: $2\pi \, \text{m}$