Statics/Geometric Properties of Lines and Areas

Centroids Of Common Shapes Of Areas And Lines

Triangular Area

$Area=\frac{b\ast h}{2}$

Quarter Circular Area

$Area=\frac{\pi r^2}{4}$

Semicircular Area

$Area=\frac{\pi r^2}{2}$

Semiparabolic Area

$Area=\frac{2ah}{3}$

Parabolic Area

$Area=\frac{4ah}{3}$

Parabolic Spandrel

$Area=\frac{ah}{3}$

Circular Sector

$Area=\alpha r^2$

Quarter Circular Arc

$Area=\frac{\pi }{2}$

Semi Circular Arc

$Area=\pi r$

Arc Of Circle

$Area=2\alpha r$

Area Moments Of Inertia of Common Geometric Shapes

Rectangle

$I_x=\frac{1}{3}b{h}^3$

$I_y=\frac{1}{3}h{b}^3$

$I_{x^{\prime }}=\frac{1}{12}b{h}^3$

$I_{y^{\prime }}=\frac{1}{12}h{b}^3$

Right Triangular Area

$I_x=\frac{1}{12}b{h}^3$

$I_y=\frac{1}{4}h{b}^3$

$I_{x^{\prime }}=\frac{1}{36}b{h}^3$

$I_{y^{\prime }}=\frac{1}{36}h{b}^3$

Triangular Area

$I_x=\frac{1}{12}b{h}^3$

$I_{x^{\prime }}=\frac{1}{36}b{h}^3$

Circular Area

$J_C=\frac{\pi r^4}{2}$

$I_{x^{\prime }}=I_{y^{\prime }}=\frac{\pi r^4}{4}$

Hollow circle

This is used for hollow cylinders where there is solid material between the outer and inner radius, but no material between the inner radius and the center, like a pipe's cross-section.

$I=\frac{\pi (r_o^4-r_i^4)}{4}$

$r_o$ is the outer radius $r_i$ is the inner radius

Semicircular Area

$I_x=I_y=\frac{1}{8}\pi r^4$

$I_{x^{\prime }}=(\frac{\pi }{8}-\frac{8}{9\pi })r^4$

$I_{y^{\prime }}=\frac{1}{8}\pi r^4$

Quarter Circle

$I_x=I_y=\frac{1}{16}\pi r^4$

$I_{x^{\prime }}=I_{y^{\prime }}=(\frac{\pi }{16}-\frac{4}{9\pi })r^4$

further reading

Statics/Geometric_Properties_of_Solids