In general: the sum of the interior angles in a n sided polygon is
A regular polygon is a polygon where the sides are all of equal length and the angles are all congruent (or of equal measure).
name of polygon  n 
degrees of the interior measure of each angle 
360 degrees divided by # in Column 2 
equilateral triangle  3 
60  360 / 3 = 120 
square  4 
90  360/4= 90 
regular pentagon  5 
3*180/5= 108 
360/5= 72 
regular hexagon  6 
4*180/6=120 
360/6= 60 
regular heptagon  7 
5*180/7 
360/7 
regular octagon  8 
6*180/8=135 
360/8 = 45 
regular dodecagon 
12 
10*180/12=1800/12=150

360/12=30 
(180  360/n) + (180  360/k) + (180
 360/p) = 360
3*180 360( 1/n+1/k+1/p)= 2*180
1*180 = 360( 1/n+1/k+1/p)
So, for example, n=3, k=4
and p= 5 is not possible since
Number of polygons around a vertex 
Equation for angle sum = 360  Equivalent Arithmetic equation  Solutions to the arithmetic equations.  
3: n , k, p  180  360/n+180  360/k+180  360/p = 360  1/n+1/k+1/p =1/2 


4: n, k, p, z  180  360/n+180  360/k+180  360/p 180  360/z = 360  1/n+1/k+1/p +1/z =2/2 =1 


5: n, k, p, z, w  180  360/n+180  360/k+180  360/p+180  360/z+180  360/w = 360  1/n+1/k+1/p +1/z+1/w =3/2 
