Moment of Area Formulas: Circles, Triangles, and Rectangles
 Moment of Area Formulas for Circles, Triangles and Rectangles
 Circle
Right: Diagram showing the relationship between the reference axis (generally the x-axis), and the parallel centroidal axis.
A = π r 2
Ic = π r 4¸ 4
x-axis tangent to circle:
x = r
Ax = π r 3
Ix = 5π r 4¸ 4
Generally, for any parallel axes:
First Moment of Area = Ax
Second Moment of Area:
Ix = Ic + Ax 2
 Semi-Circle
Right: A circle section positioned as per the upper sketch is defined in the calculator as I x-axis, the lower sketch shows I y-axis. A = π r 2¸ 2
Diameter perpendicular to x-axis,
centroidal axis = x-axis:
Ic = π r 4¸ 8
Diameter on x-axis,
centroidal axis parallel to x-axis:
Ic = r 4(9π 2 - 64) ¸ 72π
x = 4r ¸
Ax = 2r 3¸ 3
Ix = π r 4¸ 8
 Triangle
Right: Triangles with centroidal axes re-positioned with respect to the x-axis.
A = bh ¸ 2
Ic = bh 3¸ 36
Base on x-axis,
centroidal axis parallel to x-axis:
x = h ¸ 3
Ax = bh 2¸ 6
Ix = bh 3¸ 12
x-axis through vertex,
Base and centroidal axis parallel
to x-axis:
x = 2h ¸ 3
Ax = bh 2¸ 3
Ix = bh 3¸ 4
 Rectangle
Right: Rectangle with its centroidal axis revolved through angle θ.
A = bd
Ic = bd 3¸ 12
Base on x-axis,
centroidal axis parallel to x-axis:
x = d ¸ 2
Ax = bd 2¸ 2
Ix = bd 3¸ 3
Centroidal axis revolved at
an angle θ with respect to x-axis:
Let Ix = bd 3¸ 12 and Iy = b3d ¸ 12
then Irev. = Ix (cosθ)2 + Iy (sinθ)2
 Circle - Circular Sections = Triangle
Right: Diagram of integrals Ix-axis (Section B), Iy-axis (Section C), and xsinθ (Section A, dashed vertical line).
Ix-axis, Iy-axis, Ax, A, and x are evaluated for sections A, B, C and the circumscribed circle. Each section is the sum of its elemental rectangles, therefore:
Irev. = Ix (cosθ)2 + Iy (sinθ)2
From the circumcircle diameter:
Section I = Irev. ± A(xsinθ)2, and the
centroidal axes are at = ± A(xsinθ)