Moment of Inertia Calculator
Calculate moment inertia instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed by Manoj Kumar, Mathematics Educator
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\nIy = hb^3/12 = 200 x 100^3 / 12 = 200 x 1,000,000 / 12 = 16,666,667 mm^4\nArea = 100 x 200 = 20,000 mm^2\nSx = Ix / (h/2) = 66,666,667 / 100 = 666,667 mm^3\nrx = 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\nFlange area = 150 x 12 = 1,800 mm^2 each\nWeb area = 8 x 200 = 1,600 mm^2\nTotal area = 2(1,800) + 1,600 = 5,200 mm^2\nFlange Ix = 150(12)^3/12 = 21,600 mm^4\nWeb Ix = 8(200)^3/12 = 5,333,333 mm^4\nFlange d = (200+12)/2 = 106 mm\nIx = 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.
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 by Manoj Kumar, Mathematics Educator · Editorial policy