The polar moment of inertia of a hollow cylinder (j), where the inner diameter and outer diameter are known, also known as the formula for the second moment of area of a hollow cylinder, is derived from the second moment of area and results in:

, where:

J = Polar Moment of Inertia of a hollow cylinder (m^4)

= pi, 3.1415...

The polar moment of inertia of a rectangle or rectangular cross section (j), also known as the formula for the second moment of area of a rectangle, is derived from the second moment of area and results in:

, where:

J = Polar Moment of Inertia of a rectangle or rectangular section (m^4)

The polar moment of inertia of a square or square cross section (j), also known as the formula for the second moment of area of a square, is derived from the second moment of area and results in:

, where:

J = Polar Moment of Inertia of a square or square section (m^4)

b = Length or width of the square's sides (m)

, where:

= Porosity

V_{V} = Volume of Void Space

V_{T} = Total Volume of Material (Including Void)

, where:

E = Potential Energy of a Spring (J)

k = Spring Constant (N/m)

x = Distance of Compression or Expansion (m)