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

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

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

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

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