Answer:
The degree measure of the requested angle is [tex]72^o[/tex]
Step-by-step explanation:
Recall that the formula for the arc length (S) is given by:
[tex]s=R\,*\theta[/tex]
where R is the circle's radius and the angle [tex]\theta[/tex] (which is the angle subtended by the arc "S") must be given in radians.
So first, we need to find the radius of the circle, given the information on its circumference ([tex]20\,\pi \,[/tex] cm).
Since the circumference of a circle is given by: [tex]2\,\pi\,R[/tex],
then we can find what the radius is in our case:
[tex]2\,\pi\,R = 20\,\pi\,\,cm\\R=\frac{20\,\pi}{2\,\pi} \,cm\\R=10\,\,cm[/tex]
Now, with the radius (10 cm) we can use the arc length formula to find the subtended angle [tex]\theta[/tex] in radians:
[tex]s=R\,*\theta\\4\,\pi\,\,cm=(10\,\,cm)\,\theta\\\theta=\frac{4\,\pi}{10} \\\theta=\frac{2}{5} \,\pi[/tex]
Now we find the degree equivalent to this angle, via multiplying it by [tex]\frac{180^o}{\pi}[/tex], which renders:
[tex]\frac{2}{5} \pi\,(\frac{180^o}{\pi} )=72^o[/tex]
So this is the degree measure of the requested angle.
