What is the length of a diagonal of a regular pentagon of unit side length?
Sketch a regular pentagon and one of its diagonals
-- that diagonal, along with two sides forms an isosceles triangle
You know two of the sides of that triangle; they are each 1
You can find the size of the vertex angle of the pentagon by using this formula: (n - 2)·180o / n
-- since n = 5: each vertex angle = (5 - 2)·180o / 5 = 108o.
To find the length of the diagonal (the base of the isosceles triangle), use the Law of Cosines:
-- c2 = a2 + b2 - 2·a·b·cos(C)
-- c2 = 12 + 12 - 2·1·1·cos(108o)
The value of c will be the length of a diagonal of a regular pentagon of unit side length ...