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Centroids of Common Shapes

The centroid is the geometric center of an area. For composite areas, it's found by summing the moments of individual areas about an axis and dividing by the total area.

Formulas for Simple Shapes

Rectangle: $A = bh$, $\bar{x} = b/2$, $\bar{y} = h/2$
Triangle: $A = \frac{1}{2}bh$, $\bar{x} = b/3$ (from vertical leg), $\bar{y} = h/3$ (from horizontal leg)
Semicircle: $A = \frac{1}{2}\pi r^2$, $\bar{x} = r$, $\bar{y} = \frac{4r}{3\pi}$ (from diameter)
Quarter Circle: $A = \frac{1}{4}\pi r^2$, $\bar{x} = \frac{4r}{3\pi}$, $\bar{y} = \frac{4r}{3\pi}$ (from straight edges)

Centroid of Composite Area

For a composite area made of $n$ parts:

Total Area: $A_{total} = \sum A_i$
Centroid x-coordinate: $\bar{X} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$
Centroid y-coordinate: $\bar{Y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$
For holes, the area $A_i$ is negative.

Moments of Inertia (Second Moment of Area)

Moment of inertia is a measure of an object's resistance to bending or rotation about an axis.

Parallel-Axis Theorem

To find the moment of inertia $I$ about an axis parallel to the centroidal axis $I_c$:

$I_x = I_{cx} + Ad_y^2$
$I_y = I_{cy} + Ad_x^2$
Where $A$ is the area, $d_y$ is the perpendicular distance from the centroidal x-axis to the parallel x-axis, and $d_x$ is the perpendicular distance from the centroidal y-axis to the parallel y-axis.

Moments of Inertia for Simple Shapes (about their own centroidal axes)

Rectangle (base $b$, height $h$):
- $I_{cx} = \frac{1}{12}bh^3$
- $I_{cy} = \frac{1}{12}hb^3$
Triangle (base $b$, height $h$):
- $I_{cx} = \frac{1}{36}bh^3$
- $I_{cy} = \frac{1}{36}hb^3$
Circle (radius $r$):
- $I_{cx} = I_{cy} = \frac{1}{4}\pi r^4$
Semicircle (radius $r$, about diameter):
- $I_{cx} = \frac{1}{8}\pi r^4$
- $I_{cy} = (\frac{1}{8}\pi - \frac{64}{9\pi}) r^4$ (about centroidal axis)

Steps for Composite Area Calculations

Divide the composite area into simple geometric shapes (rectangles, triangles, circles). Treat holes as negative areas.
Determine the area ($A_i$) and the centroidal coordinates ($\bar{x}_i, \bar{y}_i$) for each simple shape.
Calculate the total area and the overall centroid ($\bar{X}, \bar{Y}$) of the composite shape.
Calculate the moment of inertia ($I_{cxi}, I_{cyi}$) for each simple shape about its own centroidal axes.
Apply the Parallel-Axis Theorem to find the moment of inertia of each shape about the desired axis (e.g., the composite area's centroidal axis or a specified reference axis).
- $I_{xi} = I_{cxi} + A_i d_{yi}^2$
- $I_{yi} = I_{cyi} + A_i d_{xi}^2$
Sum the moments of inertia of all individual shapes to get the total moment of inertia for the composite area. Remember to subtract for holes.

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