Express sum of side and diagonal of a nonagon.

Question

For a regular nonagon, how can I express sum of the side and the smallest
diagonal in terms of other lengths associated with nonagon?

So far I have determined that regular nongon can be inscribed into a circle.
And also if I divide it into 9 triangles angles with each side equal to the radius of circumcircle and angle becomes 40-70-70.

Which properties of regular nonagon or another polygon can be used to express sum of smallest diagonal and side?

Answer 1

"And also if I divide it into 9 triangles angles with each side equal to the radius of circumcircle and angle becomes 40-70-70."

Correct. So if we label the radius as $h$ (I'm thinking it is the hypotenuse of a right triangle of one of these isoceles trinagles cut in half) then the altitude from center to midpoint of side is $h*\sin 70$ and the side of the nonagon is $2*h*\cos 70$.

The smallest diagonal and two sides will create a triangle with angles 140-20-20. The sides are $s= 2h\cos 70$ and so the diagonal is .... what?