Centroids Areas

Overview

Engineering reference data for Centroids Areas in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Symbol	Description	Unit
$F$	Force	N
$m$	Mass	kg
$a$	Acceleration	m/s²
$v$	Velocity	m/s

Geometric Centroid Formulas

The centroid coordinates for common geometric shapes are derived from their dimensions and properties.

Square

For a square with side length a:

• Centroid x-coordinate: $x = \frac{a}{2}$
• Centroid y-coordinate: $y = \frac{a}{2}$
• Area: $A = a^2$

Rectangle

For a rectangle with sides a (width) and b (height):

• Centroid x-coordinate: $x = \frac{a}{2}$
• Centroid y-coordinate: $y = \frac{b}{2}$
• Area: $A = a \cdot b$

Circle

For a circle with radius r:

• Centroid x-coordinate: $x = r$
• Centroid y-coordinate: $y = r$
• Area: $A = \pi r^2$

Semi-Circle

For a semi-circle with radius r:

• Centroid x-coordinate: $x = \frac{4r}{3\pi}$
• Centroid y-coordinate: $y = r$
• Area: $A = \frac{\pi r^2}{2}$

Right-angled Triangle

For a right-angled triangle with base b and height h:

• Centroid x-coordinate: $x = \frac{b}{3}$
• Centroid y-coordinate: $y = \frac{h}{3}$
• Area: $A = \frac{b \cdot h}{2}$

Properties of Centroids

• The centroid is the geometric center of a shape's area, also known as the center of mass for a uniform density.
• For composite shapes, the overall centroid can be found by breaking the shape into simpler parts and calculating the weighted average of their individual centroids.
• The centroid coordinates are measured from a chosen reference point or axis.

Quick Reference: Centroid Formulas for Common Shapes

A concise summary of centroid coordinates and areas for standard geometric shapes.

Shape	x-coordinate	y-coordinate	Area
Square	$\frac{a}{2}$	$\frac{a}{2}$	$a^2$
Rectangle	$\frac{a}{2}$	$\frac{b}{2}$	$ab$
Circle	$r$	$r$	$\pi r^2$
Semi-Circle	$\frac{4r}{3\pi}$	$r$	$\frac{\pi r^2}{2}$
Right-angled Triangle	$\frac{b}{3}$	$\frac{h}{3}$	$\frac{bh}{2}$

Variables are as defined in the Variables section. Centroids are measured from the shape's reference axes.

References

• Original Source