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 = b^{3}d ¸ 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) |