Centroids Of Common Shapes Of Areas And Lines

Triangular Area

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

Quarter Circular Area

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

Semicircular Area

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

Semiparabolic Area

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

Parabolic Area

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

Parabolic Spandrel

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

Circular Sector

${\displaystyle Area=\alpha \ r^{2}}$

Quarter Circular Arc

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

Semi Circular Arc

${\displaystyle Area=\pi \ r}$

Arc Of Circle

${\displaystyle Area=2\alpha \ r}$

Area Moments Of Inertia of Common Geometric Shapes

Rectangle

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

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

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

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

Right Triangular Area

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

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

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

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

Triangular Area

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

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

Circular Area

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

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

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.

${\displaystyle I={\frac {\pi \ (r_{o}^{4}-r_{i}^{4})}{4}}}$

${\displaystyle r_{o}}$ is the outer radius ${\displaystyle r_{i}}$ is the inner radius

Semicircular Area

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

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

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

Quarter Circle

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

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