How Do Formulas for Sector Area and Arc Length Relate?

[frozonecom]

I'm just wondering because I'm really confused right now.

My teacher gave us the formula:

$K= \frac{1}{2}sr$

for area of a given sector where "s" is the arc length and "r" is the given radius.

the formula for the arc length is:

$s=\Theta r$

Though, I can't seem to understand how he came up with the formula for the area of the sector, because searching the internet always came with the result that the formula for area of the sector is:

$A= \frac{\Theta}{360} \pi r^2$

I hope someone can help me. :)

 

[chogg]

Your first formula is measured in "radians". A full revolution is 2*pi radians.

Your last formula is measured in degrees. A full revolution is 360 degrees.

Multiply your last formula by 1 in a clever way: 1 = (360 degrees)/(2*pi)

 

[frozonecom]

Oh! So both formulas are actually the same.

The difference is just that I'll use the "right formula" based on the given theta!

Thanks! :)

 

[pmsrw3]

Here's an easy way to think of it. You know that the area of a circle is pi r^2, right? So, think of the circle as a "sector" of angle 360 degrees, or 2pi radians. You need to multiply the angle by whatever factor will give you the result pi r^2. And then that factor is the same for any other angle measured in the same units.