TITLE: What length would the sides of a triangle over Earth's surface be for the sum of its angles to be 180.1°
QUESTION [9 upvotes]: For simplicity's sake, let the Earth be a perfect sphere. Imagine you are drawing an equilateral triangle over its surface. How long should its sides be, for the sum of its angles to be 180.1 degrees?

REPLY [2 votes]: Consider an equilateral spherical triangle with side lengths $a$ and angles $A$, by the supplemental cosine rule https://en.wikipedia.org/wiki/Spherical_trigonometry#Identities
\begin{eqnarray*}
\cos(A) =- \cos^2(A)+\sin^2(A) \cos(a) \\
\cos(a) =\frac{\cos(A)+\cos^2(A)}{\sin^2(A)} =\frac{\cos(A)}{1-\cos(A)}
\end{eqnarray*}
Now we have $A=60.0333\cdots$, which gives $a=3.6\cdots$. 
To obtain the distance on a sphere of radius $r$ calculate $ra\pi/180$.