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# Help

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

Guest Apr 3, 2017
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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
edited by CPhill  Apr 3, 2017
edited by CPhill  Apr 3, 2017

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