The formulas for arc length and area of a sector are quite close in their appearance. The formula for arc length, however, is related to the circumference of a circle while the area of a sector is related to, well, the area! The arc length formula is [tex]AL= \frac{ \beta }{360} *2 \pi r[/tex]. Notice the "2*pi*r" which is the circumference formula. The area of a sector is [tex]A s= \frac{ \beta }{360} * \pi r ^{2} [/tex]. Notice the "pi*r squared", which of course is the area of a circle. In our problem we are given the arc length and the radius. What we do not have that we need to then find the area of a sector of the circle is the measure of the central angle, beta. Filling in accordingly, [tex]6 \pi = \frac{ \beta }{360} *2 \pi (18)[/tex]. Simplifying by multiplying by 360 on both sides and then dividing by 36 on both sides gives us that our angle has a measure of 60°. Now we can use that to find the area of a sector of that same circle. Again, filling accordingly, [tex] A_{s} = \frac{60}{360} * \pi (18) ^{2} [/tex], and [tex] A_{s} =54 \pi [/tex]. When you multiply in the value of pi, you get that your area is 169.65 in squared.
