Area Moment of Inertia

Second Moment of Inertia or Area Moment of Inertia

Area Moment of Inertia, also known as Second Moment of Inertia - I, is a property of shape that is used to predict deflection, bending and stress in beams.


  • the "Area Moment of Inertia" is analogous to the "Polar Moment of Inertia" as a measure of a beam's ability to resist torsion - which is required to calculate the twist of a beam subjected to torque
  • the "Section Modulus" is defined as W = I / y, where I is Area Moment of Inertia and y is the distance from the neutral axis to any given fibre

Area Moment of Inertia - Imperial units

  • inches4

Area Moment of Inertia - Metric units

  • mm4
  • cm4
  • m4

Converting between units

  • 1 cm4 = 10-8 m = 104 mm
  • 1 in4 = 4.16x105 mm4 = 41.6 cm4
  • 1 cm3 = 10-6 m = 103 mm

Second Moment of Inertia

can be expressed as

Ix = ∫ y2 dA         (1)


Ix = second moment of inertia

y = the perpendicular distance from axis x to the element dA

dA = an elemental area

Area Moment of Inertia for some common Cross Sections

Solid Square Cross Section

area moment of inertia square section

Ix = b4 / 12         (2)


b = side


Iy = b4 / 12         (2b)

Solid Rectangular Cross Section

area moment of inertia rectangular section

Ix = b h3 / 12         (3)


b = width

h = height


Iy = b3 h / 12         (3b)

Solid Circular Cross Section

area moment of inertia circular section

Ix = π r4 / 4

    = π d4 / 64         (4)


r = radius

d = diameter


Iy = π r4 / 4

    = π d4 / 64         (4b)

Hollow Cylindrical Cross Section

area moment of inertia cylindrical section

Ix = π (do4 - di4) / 64         (5)


do = cylinder outside diameter

di = cylinder inside diameter


Iy = π (do4 - di4) / 64         (5b)

Related Topics

  • Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects
  • Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns

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