Exact radius of iOS icon rounded corners
ios - iPhone App Icons - Exact Radius? - Stack Overflow
From David Barnard of App Cubby:
After trying some of the answers in this post, I consulted with Louie Mantia (former Apple, Square, and Iconfactory designer) and all the answers so far on this post are wrong (or at least incomplete). Apple starts with the 57px icon and a radius of 10 then scales up or down from there. Thus you can calculate the radius for any icon size using 10/57 x new size (for example 10/57 x 114 gives 20, which is the proper radius for a 114px icon). Here is a list of the most commonly used icons, proper naming conventions, pixel dimensions, and corner radii.