Question

Which of these is the area of a sector of a circle with r = 18”, given that its arc length is 6π?

Answer 1

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 $AL= \frac{ \beta }{360} *2 \pi r$.  Notice the "2*pi*r" which is the circumference formula.  The area of a sector is $A s= \frac{ \beta }{360} * \pi r ^{2}$.  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, $6 \pi = \frac{ \beta }{360} *2 \pi (18)$.  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, $A_{s} = \frac{60}{360} * \pi (18) ^{2}$, and $A_{s} =54 \pi$.  When you multiply in the value of pi, you get that your area is 169.65 in squared.