LOk - "you "have" to buy the geo-slope software?" Don't think so. People used to do these things by hand calc methods (probably way before your time - but go to Terzaghi and Peck (1967) to see tables on how to do it.).
The lad/lass has a bank height - well, true, that is one parameter. I would hope they know the angle of the slope. LOk is correct in that knowing the soil types and properties is necessary (to a degree which I will explain later). Also the stratigraphy.
First - is the slope of a cohesionless soil? (sand and gravel) - if so, there really isn't a "critical circle" involved as it is an infinite slope problem that will govern to provide the minimum factor of safety. Now if this is an embankment sitting on a soft or firm clay - then there would be a global stability problem - not just the slope itself.
If the soil is cohesive - many years ago, someone wrote a paper that would give you a near/close estimate as to the location of the critical slip surface "focal/centre point" - that point from which the critical radius. The paper I am thinking of was from the mid to late 60s - but Huang (1980) also provided a method - (Huang, Y.H. (1980), Stability charts for effective stress analysis of nonhomogeneous embankments, Transportation Research Record, 749, pp 72-74. you will find this explained in Fang's Foundation Engineering Handbook, 2nd Ed 1991.
So, in essence, there are ways (including many charts) that can be used to find the "critical" surface (or close) of relatively homogenous slopes. Of course, determining the factor of safety is dependent on knowing the properties; complex slopes/stratigrapies notwithstanding.
