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 1927 when they were republished in Ramanujan's Collected Papers.
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). The problem stems from Ramanujan's work on modular equations of degree 3...