Question 1133757
Each side of a regular octagon is y cm long find the distance in cm between any two parallel sides of this octagon

Please try to solve without sin cos tan please
<pre>This is a REGULAR OCTAGON so all sides and all angles are congruent.
The distance between 2 parallel sides of an octagon is twice the measure of its apothem.
The APOTHEM is the ALTITUDE drawn from the APEX of the octagon to each base of the 8 isosceles triangles in the octagon.
The APOTHEM is also opposite the 60<sup>o</sup> angle of any of the 2 right triangles, formed when the altitude is drawn, and represents the longer leg too
The length of the APOTHEM/ALTITUDE/LONGER LEG (opposite the 60<sup>o</sup> angle) is the length of the SHORTER Leg * {{{matrix(1,3, sqrt(3), or, SL * sqrt(3))}}}
Since the length of one of the sides is y, the shorter leg (SL) of any one of the 2 right triangles, in this case, is {{{y/2}}},
so the length of the APOTHEM is: {{{matrix(1,3, (y/2) * sqrt(3), "=", y*sqrt(3)/2)}}}. 
Now, with the distance being twice the length of the APOTHEM, we get: {{{highlight_green(matrix(1,5, Distance, "=", 2 * (y * sqrt(3)/2), "=", y * sqrt(3)))}}}