﻿ Mechanics Map - 2D Centroid and Moment of Inertia Table

# Centroids and Area Moments of Inertia for 2D Shapes

Shape with Area and Centroid Location Shown Rectangular Area Moments of Inertia Polar Area Moments of Inertia

## Rectangle

$Area=bh$
$I_{x}=\frac{1}{12}bh^{3}$ $I_{y}=\frac{1}{12}b^{3}h$ $J_{z}=\frac{1}{12}bh(b^{2}+h^{2})$

## Right Triangle

$Area=\frac{1}{2}bh$
$I_{x}=\frac{1}{36}bh^{3}$ $I_{y}=\frac{1}{36}b^{3}h$
$I_{x'}=\frac{1}{12}bh^{3}$ $I_{y'}=\frac{1}{12}b^{3}h$

## Triangle

$Area=\frac{1}{2}bh$
$I_{x}=\frac{1}{36}bh^{3}$
$I_{x'}=\frac{1}{12}bh^{3}$

## Circle

$Area=\pi r^{2}$
$I_{x}=\frac{\pi}{4}r^{4}$ $I_{y}=\frac{\pi}{4}r^{4}$ $J_{z}=\frac{\pi}{2}r^{4}$

## Circular Annulus

$Area=\pi (r_{o}^{2}-r_{i}^{2})$
$I_{x}=\frac{\pi}{4}(r_{o}^{4}-r_{i}^{4})$ $I_{y}=\frac{\pi}{4}(r_{o}^{4}-r_{i}^{4})$ $J_{z}=\frac{\pi}{2}(r_{o}^{4}-r_{i}^{4})$

## Semicircle

$Area=\frac{\pi}{2} r^{2}$
$I_{x}=\left(\frac{\pi}{8}-\frac{8}{9\pi}\right) r^{4}$ $I_{y}=\frac{\pi}{8}r^{4}$
$I_{x'}=\frac{\pi}{8}r^{4}$
$J_{z}=\left(\frac{\pi}{4}-\frac{8}{9\pi}\right) r^{4}$

## Quarter Circle

$Area=\frac{\pi}{4} r^{2}$
$I_{x}=\left(\frac{\pi}{16}-\frac{4}{9\pi}\right) r^{4}$ $I_{y}=\left(\frac{\pi}{16}-\frac{4}{9\pi}\right) r^{4}$
$I_{x'}=\frac{\pi}{16}r^{4}$ $I_{y'}=\frac{\pi}{16}r^{4}$
$J_{z}=\left(\frac{\pi}{8}-\frac{8}{9\pi}\right) r^{4}$

## Ellipse

$Area=\pi ab$
$I_{x}=\frac{\pi}{4}ab^{3}$ $I_{y}=\frac{\pi}{4}a^{3}b$

## Circular Sector

$Area=\theta r^2$
$I_{x}=\frac{1}{4} \left(\theta - \frac{1}{2}\sin{2\theta} \right) r^{4}$

## Quarter Circle Arc

$Length=\frac{\pi}{2} r$

## Semicircle Arc

$Length=\pi r$

## Circular Arc Segment

$Length=2 \theta r$

## Parabolic Area

$Area = \frac{4}{3} ab$