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

regtangle with opened ring $I_{x} = \frac {1}{3}b h^3$

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

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

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

Right Triangular Area

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

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

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

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

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