how to find the apothem of a pentagon from the sum of the apothem and radius

[Thunderbird]

If all I know is the sum of the apothem and radius how can I find the apothem or radius length? The other one can easily be calculated one is known...

So far I have come up with:

Let Z = A (apothem) + R (radius) and we can find A from R * cos 36° so... Z = R + (R * cos 36°) but try as I might I cannot separate R from the multiplied term.

I've been working on this for quite a while and I have found many sites with answers to this questions when the perimeter is known or when the length of a side is known, or the area is known, but so far I have not found anything that answers my question.

Thanks in advance,
Gary

 

[Thunderbird]

Let apothem = 1. Then radius = 1/SIN(54)So ratio a:r = 1:1/SIN(54)Got it?

I didn't at first, but what I did see by substituting into my previous equation for R is... Z = 1/SIN(54) + [1/SIN(54) * COS(36)] = 1/SIN(54) + [COS(36)/SIN(54)] = [1 + COS(36)] / SIN(54) = 2.2361 approx.and I recognized that as SQRT(5) which I think has something to do with the Golden Ratio. So, that simplifies to A = Z/SQRT(5).Unfortunately for me, if I hadn't recognized that the square root of 5 I would not have caught on, however, I do have my answer thanks to you!Gary