If the length of one side of a regular pentagon is taken as 1.What is the perpendicular distance from one vertex to the opposite side?

Question

If the length of one side of a regular pentaon is taken as 1

Answer 1

By drawing such a pentagon, including a perpendicular from one vertex (at top of my drawing), I find that it passes through the center of the circumscribing circle. The distance from the vertex to the center is r, the radius of the circumscribing circle. Continuing along this perpendicular to the point of intersection with the opposite side, an additional distance d, I note that there's a right triangle having hypotenuse r and acute angle 54^o. (Its vertices are the circle's center, the foot of the perpendicular, and a vertex of the pentagon to the left of that foot.

 

Therefore tan 54^o = d / ½, so d = ½ tan 54^o.

In that same rt. triangle, cos 54^o = ½ ÷r so r = ½ / cos 54^o.

 

The perpendicular's length equals r + d.