Statics/Geometric Properties of Lines and Areas

Centroids Of Common Shapes Of Areas And Lines

Triangular Area

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

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

$I_{x}={\frac {1}{3}}bh^{3}$

$I_{y}={\frac {1}{3}}hb^{3}$

$I_{x'}={\frac {1}{12}}bh^{3}$

$I_{y'}={\frac {1}{12}}hb^{3}$

Right Triangular Area

$I_{x}={\frac {1}{12}}bh^{3}$

$I_{y}={\frac {1}{4}}hb^{3}$

$I_{x'}={\frac {1}{36}}bh^{3}$

$I_{y'}={\frac {1}{36}}hb^{3}$

Triangular Area

$I_{x}={\frac {1}{12}}bh^{3}$

$I_{x'}={\frac {1}{36}}bh^{3}$

Circular Area

$J_{C}={\frac {\pi \ r^{4}}{2}}$

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

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

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

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

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

Quarter Circle

$I_{x}=I_{y}={\frac {1}{16}}\pi \ r^{4}$

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

further reading

Statics/Geometric_Properties_of_Solids