Free Moment of Inertia Calculator

If you need to quickly find the second moment of area (also referred to as the area moment of inertia) for standard geometric profiles — such as circles, rectangles, or hexagons — this area moment of inertia calculator provides a straightforward solution. It includes the basic formulas and explains how to apply them correctly, whether you are interested in the moment about the x‑axis ($I_x$) or the y‑axis ($I_y$). By using these ready‑to‑use equations, you can avoid integration and obtain accurate results for structural design or mechanical analysis.

What Is the Second Moment of Area?

The second moment of area is a geometric quantity that measures how the points of a cross‑section are distributed relative to a given axis. This property is fundamental in determining a beam’s resistance to bending and deflection. Its standard unit is length raised to the fourth power (m⁴, cm⁴, or in⁴). Two values are normally considered: the moment about the horizontal x‑axis ($I_x$) and the moment about the vertical y‑axis ($I_y$). By convention, the width of a shape is associated with the x‑axis and its height with the y‑axis.

Moment of Inertia Formulas for Common Shapes

The expressions below are valid only when the reference axes pass through the centroid of the area. For the most frequent cross‑sections, these closed‑form equations eliminate the need for integration.

Rectangle (width $b$, height $h$):

$$I_x = \frac{b h^3}{12}, \qquad I_y = \frac{h b^3}{12}.$$

Circle (radius $r$):

$$I_x = I_y = \frac{\pi r^4}{4}.$$

Semicircle (radius $r$):

$$I_x = \left(\frac{\pi}{8} - \frac{8}{9\pi}\right) r^4, \qquad I_y = \frac{\pi}{8} r^4.$$

Ellipse (horizontal radius $r_x$, vertical radius $r_y$):

$$I_x = \frac{\pi}{4} r_x r_y^{\,3}, \qquad I_y = \frac{\pi}{4} r_y r_x^{\,3}.$$

Regular hexagon (side length $s$):

$$I_x = I_y = \frac{5\sqrt{3}}{16} s^4.$$

Triangle (base width $b$, height $h$, horizontal offset of the top vertex from the centroid $a$):

\begin{aligned} I_x &= \frac{b h^3}{36}, \$$6pt] I_y &= \frac{h b^3 - h a b^2 + b h a^2}{36}. \end{aligned}

The Parallel Axis Theorem

When the axis of interest does not coincide with the centroid, the parallel axis theorem must be applied. For any axis that is parallel to a centroidal axis and located a distance $d$ away, the moment of inertia becomes:

$$I = I_c + A d^2,$$

where $I_c$ is the centroidal moment of inertia, $A$ is the area of the cross‑section, and $d$ is the perpendicular separation between the two axes. This theorem is especially useful for composite sections, where each part may have its own centroid.

Worked Example: Rectangular Section

Consider a rectangle with a width of 12 cm and a height of 8 cm. Taking the centroid as the origin:

$$I_x = \frac{12 \times 8^3}{12} = 512\ \text{cm}^4,\qquad I_y = \frac{8 \times 12^3}{12} = 1152\ \text{cm}^4.$$

These results show that the cross‑section is stiffer about the y‑axis, since more material lies farther away from that axis.

Practical Applications

The area moment of inertia plays a central role in structural and mechanical engineering: it directly enters into beam deflection calculations, bending stress formulas, and stability checks. From $I$ you can also obtain the section modulus ($S = I / c$), a parameter required in many design codes. As a geometric property, the second moment of area is additive — the moment of a complex shape is the sum of the moments of its simple components, with voids subtracted. While this tool focuses on the second moment of area, complementary online calculators are available for mass moment of inertia, beam deflection, or joist sizing when those additional analyses are needed.