Question 33843
The measure of one vertex angle of a regular polygon is given by:
{{{(n-2)180/n}}} where n is the number of sides of the regular polygon...and, of course, it goes without saying, that n must be an integer >2(i.e, 3, 4, 5, 6, ...).
If the measure of one vertex angle of a regular polygon is 40 degrees, then we can write:
{{{(n-2)180/n = 40}}} Simplify and solve for n.
{{{(180n-360)/n = 40}}} Multiply both sides by n.
{{{180n - 360 = 40n}}} Add 360 to both sides.
{{{180n = 360+40n}}} Subtract 40n from both sides.
{{{140n = 360}}} Divide both sides by 140.
{{{n = 2.57}}} This is not possible since a regular polygon must have an integral numer of sides.

Conclusion:
There is no regular polygon whose vertex angle is 40 degrees.