Moment of Area Formulas
for Circles, Triangles and Rectangles |
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 |
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 ¸ 3π Ax = 2r 3¸ 3 Ix = π r 4¸ 8 |
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 |
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 |
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) |