Draw two lines from one point of the octagon to the other point directly across, creating a rectangle and two trapezoids. Then split the rectangle by the diagonal and the trapezoid from one of its base points to the opposite corner. This will split the regular octagon into 6 triangles, where all of the angles cover the interior of the polygon. A pattern exists with all polygons where the number of triangles that compose a shape will be two less than the number of sides of the shape it creates.
Subtract the number of sides of a regular octagon, eight, by two. This gives you the number of triangles that make up an octagon.
Multiply the difference by 180 to find the sum of all of the interior angles in the regular octagon. This will give you the sum of the angles of all of the triangles that compose the octagon, which is its total sum of its internal angles.
Divide the product by eight, the number of sides in an octagon, to find the value of one of the interior angles of a regular octagon. If the resulting number is 135 degrees, then you calculated it correctly.