To find the central angle of a polygon we just divide 360° by the number of sides that the polygon has, in this case, we have a pentagon, then the central angle of the polygon is:
[tex]CA=\frac{360}{5}=72[/tex]
In the figure this angle is represented like this:
If we bisect this angle, which is dividing it by 2, we get the angle θ:
[tex]\theta=\frac{72}{2}=36[/tex]
We know that the cosine of the angle θ is:
[tex]\cos (\theta)=\frac{10}{c}[/tex]
By solving for c from this ratio we get:
[tex]\begin{gathered} \cos (\theta)=\frac{10}{c} \\ c\times\cos (\theta)=\frac{10}{c}\times c \\ c\times\cos (\theta)=10\times\frac{c}{c} \\ c\times\cos (\theta)=10 \\ \frac{c\times\cos (\theta)}{\cos (\theta)}=\frac{10}{\cos (\theta)} \\ c\times\frac{\cos (\theta)}{\cos (\theta)}=\frac{10}{\cos (\theta)} \\ c=\frac{10}{\cos (\theta)}=\frac{10}{\cos (36)}\approx12.36\text{ cm} \end{gathered}[/tex]
Then, the measure of the radius equals 12.36 cm
