Moment of Area Formulas: Circles, Triangles, and Rectangles
LOGO Moment of Area Formulas
for
Circles, Triangles and Rectangles
LOGO
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
Relationship of reference axis to parallel centroidal axis.
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
Top: The semi-circle centroidal axis lies on the x-axis. Bottom: The centroidal axis is parallel to the x-axis.
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
Triangles: centroidal axes re-positioned with respect to the reference axis.
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
Rectangle axes revolved with respect to the reference axes.
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)
Circle - Circular Sections = Triangle.

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