# 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

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

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