Question 513928
One of the theorems of Euclidean geometry is that the sum of the exterior angles of any polygon is 360 degrees.  Since your problem specifies a regular polygon (one with equal length sides and angles), we can determine the number of angles (and therefore the number of sides) by dividing 360 degrees by 45 degrees.

{{{ 360/45 = 8 }}}

So your regular polygon is eight-sided, i.e., an octagon.

The other important theorem we need to use here is that the interior and exterior angles are supplementary angles: in other words, their sum is 180 degrees.  Since each exterior angle is 45 degrees, each interior angle is therefore 180 - 45 = 135 degrees.  Imagine an octagon (like a stop sign) and see if this makes sense.

There are eight interior angles, each of 135 degrees.  Therefore the sum of the interior angles is:

{{{ 8 * 135 = 1080 }}} degrees

In general, the sum of the interior angles of a polygon with n sides is:

{{{ (n - 2) * 180 }}} degrees

Thus the interior angles of a triangle sum to 180 degrees, the interior angles of a quadrilateral sum to 360 degrees, and so on.  So you can get the same answer as above using the formula above with n = 8.