There is a survey article by Berndt, Choi, and Kang devoted to the set of 58 Ramanujan's problems. Many of the problems have generated serious research activity since they were republished in Ramanujan's Collected Papers in 1927.
An elementary solution to the specific geometric problem you've mentioned can be found in Ramanujan's Notebooks, Part III by Berndt (Springer, 1991, pp. 244-246).