The two functions I'm using I'm trying to find the root of the root function when some other variable is subtracted. I.e. FindRoot[root[theta, r, h]-H, theta]

Whenever I do this, FindRoot outputs a lot of error messages and then continues on to output a very close approximation to the root of the function. Can anyone shed any light onto why this is for me?

Error messages look like so: Errors

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    $\begingroup$ People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. $\endgroup$ – user9660 May 28 '16 at 16:36
f[θ_, r_, x_, h_] = 
  1/2 x (r^2 ArcCos[(r - x Cot[θ])/r] + (x Cot[θ] - r) Sqrt[
        x Cot[θ] (2 r - x Cot[θ])]) - π r^2 h;

Functions which use numeric techniques should have their arguments restricted to numeric values.

root[θ_?NumericQ, r_?NumericQ, h_?NumericQ] := 
 x /. FindRoot[f[θ, r, x, h], {x, 2}]

FindRoot[root[θ, 3.7, 2.6] - 10.7, {θ, 1}]

(*  {θ -> 1.24469}  *)

root[θ /. %, 3.7, 2.6] - 10.7

(*  3.55271*10^-15  *)

FindRoot[root[θ, 4, 2.2] - 10.7, {θ, 1}]

(*  {θ -> 1.25971}  *)

root[θ /. %, 4, 2.2] - 10.7

(*  -3.55271*10^-15  *)

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