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Moment Of Inertia Calculator

Shape Type:

Axis:

Moment of Inertia: Measure of an object’s resistance to rotational acceleration about an axis. Units: length⁴ (e.g., mm⁴, cm⁴, m⁴)

The moment of inertia is a critical property in structural engineering and mechanics, measuring how resistant a shape is to bending and rotational acceleration about an axis. Whether you’re designing beams, shafts, or other structural elements, knowing the moment of inertia helps you predict how the object will behave under load.

Our Moment of Inertia Calculator simplifies these calculations by letting you input shape dimensions and select the axis of interest. It supports common shapes such as rectangles, circles, triangles, hollow circles, I-beams, and hollow rectangles, giving you accurate results instantly.

How to Use the Moment of Inertia Calculator

Select the Shape Type: Choose your object's cross-sectional shape from the dropdown menu.
Input Dimensions: Depending on the shape, enter the required dimensions such as width, height, radius, base, flange thickness, or web thickness.
Choose the Axis: Pick the axis about which you want to calculate the moment of inertia—X-axis (horizontal), Y-axis (vertical), or both.
Calculate: Press the "Calculate" button to view the moment of inertia values.
Review Results: The calculator shows the moments of inertia (Ix and Iy), cross-sectional area, radius of gyration, and the formula used.

Example Calculation

Suppose you want to calculate the moment of inertia of a rectangle with:

Width $b = 0.3 \, m$ b=0.3m
Height $h = 0.5 \, m$ h=0.5m
Axis: Both X and Y

Using the formulas:

$I_x = \frac{b h^3}{12} = \frac{0.3 \times (0.5)^3}{12} = 0.003125 \, m^4$ Ix=12bh3=120.3×(0.5)3=0.003125m4
$I_y = \frac{h b^3}{12} = \frac{0.5 \times (0.3)^3}{12} = 0.001125 \, m^4$ Iy=12hb3=120.5×(0.3)3=0.001125m4

Area $A = b \times h = 0.15 \, m^2$ A=b×h=0.15m2

Radius of gyration $r_x = \sqrt{\frac{I_x}{A}}$ rx=AIx, $r_y = \sqrt{\frac{I_y}{A}}$ ry=AIy

The calculator outputs these values with precision, helping you understand the distribution of material relative to the axes.

Supported Shapes and Formulas

Rectangle:
$I_x = \frac{b h^3}{12}$ Ix=12bh3, $I_y = \frac{h b^3}{12}$ Iy=12hb3
Circle:
$I_x = I_y = \frac{\pi r^4}{4}$ Ix=Iy=4πr4
Triangle:
$I_x = \frac{b h^3}{36}$ Ix=36bh3, $I_y = \frac{h b^3}{36}$ Iy=36hb3
Hollow Circle:
$I_x = I_y = \frac{\pi (R^4 - r^4)}{4}$ Ix=Iy=4π(R4−r4)
I-Beam:
$I_x = \frac{b h^3}{12} - \frac{(b - t_w)(h - 2 t_f)^3}{12}$ Ix=12bh3−12(b−tw)(h−2tf)3
$I_y = \frac{2 t_f b^3}{12} + \frac{(h - 2 t_f) t_w^3}{12}$ Iy=122tfb3+12(h−2tf)tw3
Hollow Rectangle:
$I_x = \frac{B H^3 - b h^3}{12}$ Ix=12BH3−bh3, $I_y = \frac{H B^3 - h b^3}{12}$ Iy=12HB3−hb3

Frequently Asked Questions (FAQs)

1. What is the moment of inertia?
It is a geometrical property that measures an object's resistance to bending or rotation about an axis.

2. Why are there two moments of inertia (Ix and Iy)?
They represent resistance to bending about two perpendicular axes—horizontal (X) and vertical (Y).

3. What units are used?
Units depend on your input dimensions but are typically length to the fourth power (e.g., $mm^4$ mm4, $cm^4$ cm4, $m^4$ m4).

4. How does the shape affect the moment of inertia?
The shape and how material is distributed significantly affect the moment of inertia, impacting structural strength.

5. Can this calculator handle composite shapes?
It handles standard shapes individually. For composite sections, you may need to calculate each part separately and apply the parallel axis theorem.

6. What is the radius of gyration?
It is a measure of how far from the axis the area is distributed, calculated as $r = \sqrt{\frac{I}{A}}$ r=AI.

7. What happens if I select “Both Axes”?
The calculator shows moments of inertia for both X and Y axes simultaneously.

8. Why does the calculator alert invalid inputs?
To ensure accurate results, inputs must be positive numbers and physically valid (e.g., inner radius < outer radius).

9. How is an I-beam’s moment of inertia calculated?
By subtracting the hollow web section from the flange sections, reflecting the beam's cross-sectional geometry.

10. Can I reset the calculator?
Yes, use the "Reset" button to clear all inputs and results.

Conclusion

The Moment of Inertia Calculator is an essential tool for engineers, architects, students, and anyone working with structural elements. It allows you to quickly determine how a shape resists bending and rotation by providing detailed, accurate results based on your input dimensions and selected axis.

By supporting multiple common shapes and offering clear formulas and calculated properties, this tool enhances understanding and aids in designing safer, more efficient structures. Use it to save time, avoid manual errors, and gain confidence in your structural calculations.