Question

The radius of a circle is 6cm. The arc length of the sector is 5π cm.

What is the measure of the central angle in radians? (write the coefficient as a faction "n/d" without spaces where n is the value of the numerator and d is the value of the denominator. For example: One half would be written as 1/2)

What is the area of the sector intercepted by the arc?

Answer 1

First we want to find the measure of the central angle in radians
The arc length formula is \(s=r(\theta)\) where s is arc length, r is radius, theta is central angle in radians
So plugging it in we have \(5\pi=6(\theta)\)
So then just solve for theta, but write the answer in the form of a fraction

Answer 2

So the central angle would be \[5 \pi/6\] right?

Answer 3

Then the area of a sector is \(A=\frac{1}{2}r^2(\theta)\)
So plug the theta in from the above, then plug in radius, then find the area

Answer 4

And then the area of the sector would be 16π?

Answer 5

\(\frac{1}{2}(6^2)(\frac{5\pi}{6}) = \frac{1}{2}(36)(\frac{5\pi}{6})=18(\frac{5\pi}{6}) = 3(5\pi)=15\pi\)