Let's solve each part step-by-step.

Part a: Area of the Semicircle

The formula for the area of a full circle is: $\text{Area} = \pi r^2$ Since we have a semicircle (half of a circle), we need to divide this by 2. Given that the radius $r$ is 9 cm:

1. Calculate the area of the full circle: $\pi r^2 = \pi (9)^2 = 81\pi$

2. Since it's a semicircle, divide by 2: $\text{Area of semicircle} = \frac{81\pi}{2} = 40.5\pi$

So, the area of the semicircle is $40.5\pi$ cm².

Part b: Area of the Sector

The sector has an arc length and radius, but we need to determine the angle $\theta$ to find the area. Here’s how we can proceed:

1. Using the Arc Length: The arc length $s$ of a sector is given by: $s = r\theta$ where $s = 40$ cm and $r = 20$ cm. Solving for $\theta$: $\theta = \frac{s}{r} = \frac{40}{20} = 2 \text{ radians}$

2. Calculating the Sector Area: The area $A$ of a sector with angle $\theta$ (in radians) is given by: $A = \frac{1}{2} r^2 \theta$ Plugging in $r = 20$ cm and $\theta = 2$ radians: $A = \frac{1}{2} \times 20^2 \times 2 = \frac{1}{2} \times 400 \times 2 = 400 \text{ cm}^2$

So, the area of the sector is 400 cm².

If you need any further clarification, let me know!

Additional Practice Questions:

1. How would you find the area of a quarter circle with radius 9 cm?
2. What would the area of a sector be if the radius is doubled but the angle stays the same?
3. How would you calculate the circumference of a semicircle with a given radius?
4. If the arc length of a sector was halved, what would happen to the angle $\theta$?
5. How do you calculate the perimeter of a semicircle?

Quick Tip:

For any sector calculations, remember that the angle $\theta$ must be in radians when applying formulas involving arc length and area.