Compute centroidal \(I_x\) and \(I_y\) for common shapes. Pick a shape, enter its dimensions, then press Calculate.
\( \text{Rectangle: } I_x=\tfrac{bh^{3}}{12},\; I_y=\tfrac{b^{3}h}{12}. \)
\( \text{Circle: } I_x=I_y=\tfrac{\pi}{4}r^{4}. \)
\( \text{Annulus: } I_x=I_y=\tfrac{\pi}{4}(R^{4}-r^{4}). \)
\( \text{Ellipse: } I_x=\tfrac{\pi}{4}ab^{3},\; I_y=\tfrac{\pi}{4}a^{3}b. \)
A Moment of Inertia Calculator evaluates how area or mass is distributed relative to an axis, quantifying resistance to bending (area MOI) or angular acceleration (mass MOI). For mass, the integral weights distance squared from a rotation axis; for planar sections, area MOI describes stiffness in beam theory. The calculator accepts common shapes (rectangles, circles, annuli, triangles, thin rods, disks) and arbitrary composites, returns centroidal values, applies the parallel-axis theorem to shifted axes, and reports polar moments for torsion. It also computes products of inertia and principal axes, which diagonalize the inertia matrix and remove coupling terms for clean design insight.
Mass MOI about axis \(k\) uses
\[ I_k=\int r_k^2\,dm. \]
Area MOI about Cartesian axes is
\[ I_x=\iint y^2\,dA,\qquad I_y=\iint x^2\,dA,\qquad I_{xy}=\iint xy\,dA. \]
The polar area moment about a point \(O\) is
\[ J_O = I_x + I_y. \]
Parallel-axis theorems shift centroidal values to any axis:
\[ I_A = I_C + A d^2,\qquad I_A = I_G + M d^2. \]
Rotating axes by \(\theta\) uses the transformation laws:
\[ I_{x'}=\frac{I_x+I_y}{2}+\frac{I_x-I_y}{2}\cos 2\theta- I_{xy}\sin 2\theta,\quad I_{y'}=\frac{I_x+I_y}{2}-\frac{I_x-I_y}{2}\cos 2\theta+ I_{xy}\sin 2\theta, \]
\[ I_{x'y'}=-\frac{I_x-I_y}{2}\sin 2\theta+ I_{xy}\cos 2\theta,\qquad \tan 2\theta_p=\frac{2I_{xy}}{I_x-I_y}\ (\text{principal axes}). \]
Rectangle \(b\times h\), centroidal: \[ I_x=\frac{b h^3}{12},\qquad I_y=\frac{h b^3}{12}. \]
Circle (radius \(R\), centroidal): \[ I_x=I_y=\frac{\pi R^4}{4},\qquad J=\frac{\pi R^4}{2}. \]
Annulus \(R_o,R_i\): \[ I_x=I_y=\frac{\pi}{4}\left(R_o^4-R_i^4\right),\qquad J=\frac{\pi}{2}\left(R_o^4-R_i^4\right). \]
Thin rod (mass \(M\), length \(L\)): \[ I_{\text{center}}=\frac{1}{12}ML^2,\qquad I_{\text{end}}=\frac{1}{3}ML^2. \]
Solid disk (mass \(M\), radius \(R\), about center): \[ I=\frac{1}{2}MR^2. \]
\(b=0.20\,\mathrm{m},\ h=0.30\,\mathrm{m}\). Centroidal \(I_x=\frac{b h^3}{12}=0.00045\,\mathrm{m}^4\). Shift to base: \(d=h/2=0.15\,\mathrm{m}\), area \(A=bh=0.06\,\mathrm{m}^2\). \[ I_{\text{base}}=I_x + A d^2 = 0.00045 + 0.06(0.15)^2 = 0.00180\,\mathrm{m}^4. \]
\(M=2\,\mathrm{kg},\ R=0.10\,\mathrm{m}\). \[ I=\frac{1}{2}MR^2=\frac{1}{2}\cdot 2\cdot(0.10)^2=0.01\,\mathrm{kg\,m}^2. \]
\(R_o=0.05\,\mathrm{m},\ R_i=0.04\,\mathrm{m}\). \[ J=\frac{\pi}{2}\left(R_o^4-R_i^4\right) \approx \frac{\pi}{2}\big(6.25{\times}10^{-6}-2.56{\times}10^{-6}\big) \approx 5.80{\times}10^{-6}\,\mathrm{m}^4. \]
Area MOI (\(\mathrm{m}^4\)) relates to bending/torsion of sections; mass MOI (\(\mathrm{kg\,m}^2\)) relates to rotational dynamics.
When the desired axis is offset by distance \(d\) from a known centroidal axis: add \(A d^2\) (or \(M d^2\)).
Yes. It sums each part’s centroidal inertia plus shift terms to obtain the total about the target axis.
Axes where \(I_{xy}=0\). They align with maximum/minimum bending resistance and simplify analysis.
Sign depends on axis orientation and area distribution. Rotating to principal axes removes the coupling term.
Stay consistent. For area MOI use length to the fourth power; for mass MOI use mass·length squared.