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Let $n$ equal the number of sides in a regular polygon. For \(3\leq n <10\), how many values of $n$ result in a regular polygon where the common degree measure of the interior angles is not an integer?

 Jun 27, 2018
 #1
avatar+98182 
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Note that the  "formula"  for determining the measure of an interior angle in any regular polygon  is

 

180 (n - 2) / n

 

Note that an integer value for an interior angle's measure will occur  whenever  n  divides 180  evenly

 

So regular polygons  with sides of 3,4,5,6 and 9  will have integer-valued interior angle measures

 

Also..... when n  = 8, the interior angle will measure  180 * 3/4  = 135°

 

So.... only the polygon with a side of 7 will not have an interior angle with an integer-valued measure

 

 

cool cool cool

 Jun 27, 2018
 #2
avatar+21860 
0

Let $n$ equal the number of sides in a regular polygon.
For \( 3\leq n <10\),
how many values of $n$ result in a regular polygon
where the common degree measure of the interior angles is not an integer?

 

Formula:

\(\begin{array}{lcll} \text{interior angle $= 180^{\circ} - $ exterior angle $\\$ }\\ \text{exterior angle $= \dfrac{360^{\circ} }{n} \\$ }\\ \boxed{\text{interior angle}= 180^{\circ} - \dfrac{360^{\circ} }{n} }\\ \end{array} \)

 

Note that an integer value for an interior angle's measure will occur  whenever  n  divides \(360^{\circ} \) evenly

The divisors of 360 are:

\(\text{$1 | 2 | {\color{red}3} | {\color{red}4} | {\color{red}5} | {\color{red}6} | {\color{red}8} | {\color{red}9} | 10 | 12 | 15 | 18 | 20 | 24 | 30 | 36 | 40 | 45 | 60 | 72 | 90 | 120 | 180 | 360 (24 $ divisors ) }\)

 

Only Number 7 is not a divisor of 360,

so only the polygon with a side of 7 will not have an interior angle with an integer-valued measure

 

laugh

 Jun 28, 2018

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