Slenderness ratio is a value used by mechanical engineers to help determine the strength of various types of columns. Specifically, slenderness ratio is used to determine buckling stresses. It is defined as the effective length divided by the radius of gyration. Effective length is determined by the types of connections that are holding the column in place, since different connection types will resist buckling better than others, and radius of gyration depends on the column moment of inertia and its crass sectional area.
Determine the effective length by multiplying the "effective length constant," K, by the length of the column. This constant depends on how the two ends of the column are supported. Most columns have fixed/fixed connections where K = 0.5. Other possibilities are pinned/pinned, pinned/fixed and free/fixed with K equal to 1.0, 0.699 and 2.0, respectively.
Calculate the moment of inertia, I(xx). The moment of inertia about the x axis will almost always be the one you need, as long as standard conventions are being used. The formula you use will be dependent on the geometry of your cross section, or if you are using a standard cross section, you can look up the value in an engineering table.
Calculate the cross section area for your column. For standard beam and column geometries, this value can also be found in engineering tables.
Multiply the moment of inertia by the cross section area and divide the effective length by the result to determine the slenderness ratio.
A fixed joint can neither translate--move side-to-side--nor rotate. A pinned joint can not translate, but it can rotate. A free end can translate and rotate.