Moment of Inertia Calculator
Calculate moment inertia instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Moment of Inertia Calculator
Calculator
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Formula: Ix = bh^3/12 (rectangle) | I = pi*r^4/4 (circle)
Additional inputs: Web Thickness (mm).
Worked example — Ix = 66.667 x 10^6 mm^4 | Iy = 16.667 x 10^6 mm^4 | Sx = 666,667 mm^3
Formula
Ix = bh^3/12 (rectangle) | I = pi*r^4/4 (circle)
The moment of inertia depends on the cross-sectional shape. For composite sections, the parallel axis theorem (I = I_centroid + Ad^2) is used. Section modulus S = I/c where c is distance to extreme fiber. Radius of gyration r = sqrt(I/A).
Worked Examples
Example 1: Rectangular Beam Cross-Section
Problem:Calculate the moment of inertia for a rectangular beam with width 100 mm and height 200 mm.
Solution:Ix = bh^3/12 = 100 x 200^3 / 12 = 100 x 8,000,000 / 12 = 66,666,667 mm^4 Iy = hb^3/12 = 200 x 100^3 / 12 = 200 x 1,000,000 / 12 = 16,666,667 mm^4 Area = 100 x 200 = 20,000 mm^2 Sx = Ix / (h/2) = 66,666,667 / 100 = 666,667 mm^3 rx = sqrt(Ix/A) = sqrt(66,666,667/20,000) = 57.74 mm
Result:Ix = 66.667 x 10^6 mm^4 | Iy = 16.667 x 10^6 mm^4 | Sx = 666,667 mm^3
Example 2: I-Beam Cross-Section
Problem:Calculate the moment of inertia for an I-beam with 150 mm flange width, 12 mm flange thickness, 200 mm web height, and 8 mm web thickness.
Solution:Total height = 200 + 2(12) = 224 mm Flange area = 150 x 12 = 1,800 mm^2 each Web area = 8 x 200 = 1,600 mm^2 Total area = 2(1,800) + 1,600 = 5,200 mm^2 Flange Ix = 150(12)^3/12 = 21,600 mm^4 Web Ix = 8(200)^3/12 = 5,333,333 mm^4 Flange d = (200+12)/2 = 106 mm Ix = 2(21,600 + 1,800 x 106^2) + 5,333,333 = 45,820,133 mm^4
Result:Ix = 45.820 x 10^6 mm^4 | Area = 5,200 mm^2 | Efficient bending resistance
Frequently Asked Questions
What is moment of inertia and why is it important?
Moment of inertia, also called the second moment of area, is a geometric property that quantifies how a cross-sectional shape resists bending about a particular axis. It is measured in units of length to the fourth power (mm^4 or in^4). The moment of inertia is critically important in structural engineering because it directly determines how much a beam will deflect under load and the maximum stress it will experience. A larger moment of inertia means greater resistance to bending, which is why structural shapes like I-beams are designed to maximize this property by placing material far from the neutral axis. Understanding moment of inertia is essential for designing safe and efficient structural members in buildings, bridges, machinery, and all load-bearing applications.
What is the difference between Ix and Iy?
Ix and Iy represent the moment of inertia about the x-axis (horizontal) and y-axis (vertical) respectively, both measured through the centroid of the cross-section. For most structural applications, Ix (the strong axis) determines resistance to vertical bending loads, while Iy (the weak axis) determines resistance to lateral bending. The distinction is crucial because most cross-sections are not symmetric about both axes, resulting in different stiffness properties in different directions. For example, a rectangle oriented with its long dimension vertical has a much larger Ix than Iy, meaning it resists vertical bending much better than horizontal bending. This is why beams are typically oriented with their strong axis aligned with the primary loading direction to maximize structural efficiency.
What is the section modulus and how is it used?
The section modulus (S) is derived from the moment of inertia by dividing it by the distance from the neutral axis to the extreme fiber of the cross-section (S = I/c). It is measured in units of length cubed (mm^3 or in^3). The elastic section modulus is used directly in the flexure formula (stress = M/S) to calculate the maximum bending stress in a beam under a given moment. A larger section modulus means lower bending stress for the same applied moment. The plastic section modulus (Z) represents the shape factor for fully plastic conditions and is used in plastic design methods. Structural steel design codes use the section modulus extensively for sizing beams, checking capacity, and ensuring adequate safety margins against bending failure.
What is the parallel axis theorem and when do I use it?
The parallel axis theorem, also known as Steiner's theorem, states that the moment of inertia about any axis parallel to an axis through the centroid equals the centroidal moment of inertia plus the product of the area and the square of the distance between the two axes (I = I_centroid + A*d^2). This theorem is essential when calculating the moment of inertia of composite cross-sections made up of simpler shapes. For example, an I-beam can be decomposed into three rectangles (two flanges and a web), and the parallel axis theorem is used to transfer each rectangle's own moment of inertia to the common centroidal axis of the entire section. Without this theorem, it would be extremely difficult to calculate properties of complex cross-sections that are standard in structural engineering practice.
How does the radius of gyration relate to moment of inertia?
The radius of gyration (r) is defined as the square root of the moment of inertia divided by the cross-sectional area (r = sqrt(I/A)). It represents the distance from the axis at which the entire area could be concentrated to produce the same moment of inertia. The radius of gyration is particularly important in column design because it directly relates to the slenderness ratio (L/r), which determines whether a column will fail by material yielding or by elastic buckling. A larger radius of gyration means a more efficient cross-section for resisting buckling. For columns, designers check the slenderness ratio about both the strong and weak axes, and the critical case is typically the weak axis where the radius of gyration is smallest. This is why wide-flange sections are preferred for columns over narrow deep sections.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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