If a regular pentagon has sides of four feet, what is the length from the vertex to the midpoint of the opposite side
Draw the 5 equal isosceles triangles and consider one of them.
the base angles are 54° each, and we want the height of the triangle.
sin 54° = height /2
height = 2sin54° = appr 1.618
note: this is (1 + √5)/2 , which is the Golden Ratio.
If you google, "Golden Ratio" or "Fibonacci numbers" you will be kept busy for a lifetime of real fun math.