Find the radius of a circle circumscribed in a regular pentagon whose perimeter is 28 inches.

Guest Apr 3, 2017

#1**+2 **

I think you must mean "inscribed" if the circle is* inside* the regular pentagon...

Look at this picture to get a good idea of the situation :

AE = 28/5 inches = 5.6 inches....so AM is 1/2 of this = 2.8 inches

Angle AFE = 360/5 = 72°.....so.....angle AFM is 1/2 of this = 36°

And angle BAE = 108°.....so......angle FAM is 1/2 of this = 54°

And FM is the radius of the circle which can be found using The Law of Sines

AM / sin AFM = FM / sin FAM

2.8 / sin 36° = FM / sin 54°

Rearrange as

FM = sin 54° * 2.8 / sin 36° ≈ 3.854 inches

CPhill
Apr 3, 2017