Question 74773
The reason why radians are used is because with a unit circle the angle measure is the arc length. So if I have an angle with the measure of pi, then I have an arc length of pi. So the arc length formula of a unit circle is
*[Tex \LARGE s=\theta]

So if we have a circle with a radius of 2, we just multiply the formula by 2. 
*[Tex \LARGE s=2\theta]
For a general radius r we have
*[Tex \LARGE s=r\theta]
So for a radius of 2 cm and an angle of *[Tex \LARGE \theta=\frac{9\pi}{20}] the arc length is:
*[Tex \LARGE s=(2)(\frac{9\pi}{20})]
*[Tex \LARGE s=\frac{9\pi}{10}]
So the arc length is *[Tex \LARGE \frac{9\pi}{10}]





To find the area of a circle, the formula is given as:
*[Tex \LARGE A=\pi r^2]
So to find the area of a sector, we just multiply this total area by a fraction. For instance, if we have half a circle (with an angle of pi) we multiply the total area by 1/2. 
*[Tex \LARGE \frac{\not\pi}{2\not\pi}=\frac{1}{2}]
*[Tex \LARGE \frac{A}{2}=\frac{\not\pi}{2\not\pi} \pi r^2]
*[Tex \LARGE \frac{A}{2}=\frac{1}{2} \pi r^2]
For any general *[Tex \LARGE \theta] we do the same thing
*[Tex \LARGE A=\frac{\theta}{2\not\pi} \not\pi r^2]
*[Tex \LARGE A=\frac{\theta}{2} r^2]
So the area of a sector is 
*[Tex \LARGE A=\frac{1}{2} r^2 \theta]
So lets plug in r=2 and *[Tex \LARGE \theta=\frac{9\pi}{20}]
*[Tex \LARGE A=\frac{1}{2} (2)^2(\frac{9\pi}{20})]
*[Tex \LARGE A=\frac{9\pi}{10}]
So the area is *[Tex \LARGE \frac{9\pi}{10}]