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# Geometry question please explain

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Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.

A square, a regular pentagon, and a regular \$n\$-gon, all with the same side length, also completely surround a point. Find \$n\$.

May 11, 2020

### 4+0 Answers

#1
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May 11, 2020
edited by Dragan  May 16, 2020
edited by Dragan  May 16, 2020
#2
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The formula to find the number of degrees in each vertex angle of a regular polygon:

degrees  =  (n - 2)(180)/n

There are 360o around each point.

Each vertex angle of a square has 90o.

Each vertex angle of a regular pentagon has 108o.                   [ (5-2)(180)/5  =  108 ]

There are  360o - 90o - 108o  =  162o  remaining.

Using the formula:  (n - 2)(180)/n  =  162

(n - 2)(180)  =  162n

180n - 360  =  162n

-360  =  -18n

20  =  n

It's a 20-sided regular polygon.

May 11, 2020
#3
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That answer is wrong.

Guest May 12, 2020
#4
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It is correct!!!

Dragan  May 16, 2020