where G is found from the material's elastic modulus E and Poisson ratio n,

and D is the mean diameter of the spring (measured from the centers of the wire cross-sections),

The rise angle of the spring coils (the angle between the coils and the base of the spring) is found from the arctangent of the coil-pitch divided by the spring circumference,

The solid height of the spring is found by summing the widths of all the spring coils. The total number of spring coils is equal to the active coils in the spring interior plus the 2 coils at the spring ends (more),

The length of wire needed to make the spring is found from,

The maximum shear stress in the spring associated with the maximum force is given by,

where W is the Wahl correction factor (accounting for spring curvature stress) and C is the spring index (essentially an aspect ratio of the spring cross-section),

where k is the spring constant from above and M is the spring mass (see derivation). The spring mass M can be found by weighing the spring, or by finding the spring volume and multiplying by its material density,

We can express the spring's lowest resonance in terms of basic spring geometry if we substitute for k and M in the equation for f_res (and then eliminate L_wire). Doing so gives,

For springs with small rise angles and several active coils we can make the approximation,

If we also allow the approximation,

we can then simplify the resonant frequency formula to a form that can be found in several reference books,