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?
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
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