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?

Guest Jun 27, 2018

#1**+1 **

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

CPhill Jun 27, 2018

#2**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

heureka Jun 28, 2018