Moment Of Inertia - Ix, Iy, and J in m^4

The moment of inertia calculator returns the centroidal second moment of area Ix and Iy in m^4 for six common cross-sections, plus the polar J = Ix + Iy. Pick a triangle, rectangle, circle, semicircle, ellipse, or regular hexagon, type the size, and read the closed-form result in the same unit raised to the fourth power.

• • Beam and shaft design: Pull Ix and Iy for a rectangular bar, solid round shaft, or hexagonal stand-in so bending stress and torsion checks can be read off the same number.
• • Cross-section comparison: See how much stiffer a hexagon or a circle is than a rectangle of the same bounding dimension, without rederiving the inertia by hand.
• • Sensitivity tests: Re-enter one dimension to see how rapidly the centroidal inertia scales with the third or fourth power of size, the rule of thumb behind beam depth changes.

Every cross-section has its own closed-form expression, a shape constant multiplied by the linear dimension raised to the third or fourth power. The same form returns Ix and Iy for all six shapes in m^4 without changing units.

Centroidal means the formulas assume the x and y axes both pass through the geometric centroid, the point where the area would balance on a pin. For a different axis, apply the parallel-axis theorem I_parallel = I_centroidal + A * d^2 and use the result in the next beam or shaft equation.

The shape selector is the same kind used in the Cross Sectional Area Calculator, so any cross-section area you already trust for weight or paint can be paired with the matching Ix and Iy here.

The moment of inertia calculator reads the shape selector and the dimension fields, validates the inputs, and returns the centroidal Ix and Iy from the closed-form expression for the selected shape. The same step also computes the cross-section area in m^2 and the polar J = Ix + Iy in m^4, so the next calculation in a beam or shaft design has every number it needs.

• b, a (rectangle or triangle): Width and height in metres. Ignored for circle, semicircle, ellipse, and hexagon.
• r (circle or semicircle): Radius in metres. Ignored for every other shape.
• a, b (ellipse): Horizontal and vertical semi-axes in metres. Only used when the ellipse is selected.
• s (hexagon): Side length in metres. Only used when the hexagon is selected.

For a circle, the closed-form expression collapses to Ix = Iy = pi * r^4 / 4, so the polar J = pi * r^4 / 2. This is the textbook shaft design formula for a solid round bar.

According to Omni Calculator, the second moment of area of a rectangle of width b and height a about the centroidal x-axis is b * a^3 / 12, and the same family of closed-form formulas covers triangle, circle, semicircle, ellipse, and regular hexagon.

The Ix and Iy returned here are exactly the inputs the Beam Deflection Calculator expects, so a rectangular bar sized in this tool can be dropped into a deflection calculation without unit changes.

Four terms show up in every cross-section problem. Keeping them straight is the difference between a clean answer and a wrong one by a factor of four.

When Ix and Iy match, the shape has at least one axis of symmetry about both axes, which is the case for the circle, the regular hexagon, the square, and a symmetric rectangle. For an asymmetric triangle with a non-zero top-vertex offset, the two answers diverge and the difference tells a structural engineer how much the bar will twist under an off-centre load.

The bending stress sigma = M * c / I uses the same Ix or Iy this tool returns, so the Bending Stress Calculator is the natural next step once the cross-section is fixed.

Pick a cross-section, type the matching dimensions in any linear unit, and read Ix, Iy, J, and the cross-section area in the same unit raised to the fourth power or squared.

1. 1 Choose the cross-section: Open the Cross-Section Shape menu and pick triangle, rectangle, circle, semicircle, ellipse, or regular hexagon. The closed-form formula switches to match the choice.
2. 2 Enter the matching dimensions: Type width and height for a rectangle or triangle, radius for a circle or semicircle, the two semi-axes for an ellipse, or a single side length for the hexagon.
3. 3 Adjust the triangle offset if needed: For a triangle, slide the top-vertex offset between -width/2 and width/2. Zero keeps the triangle symmetric and gives the textbook Ix and Iy.
4. 4 Read Ix, Iy, J, and the area: The result row prints Ix, Iy, J, and the cross-section area. Ix and Iy are the centroidal second moments, J = Ix + Iy is the polar moment for torsion.
5. 5 Reuse the result in a beam or shaft calculation: Take Ix and Iy straight into a bending stress, beam deflection, or shaft torsion equation. The result is already in m^4 as long as the input was in metres.

A 50 mm by 25 mm steel bar for a small bracket: pick rectangle, enter 0.05 for width and 0.025 for height, and read Ix = 6.51e-8 m^4 and Iy = 2.6e-7 m^4. The same numbers feed the next beam check.

The Ix and Iy just printed in the result row are the same dimensions a beam check reads, so the Beam Load Calculator picks up the same bar and returns the supported load without retyping the size.

Reading the centroidal Ix and Iy for six common cross-sections in one tool keeps beam sizing, shaft design, and textbook checks on a single page.

• • Six cross-sections, one tool: Cover the most common centroidal Ix and Iy problems in a single calculator so a switch from a rectangle to a circle or a hexagon does not require a new tool or unit system.
• • Centroidal Ix and Iy together: Return both Ix and Iy in the same response so a comparison between the two axes does not require a second tool.
• • Polar J without retyping: Add Ix and Iy automatically and show the polar J as a derived row, so a shaft torsion calculation does not have to repeat the addition by hand.
• • Visible closed-form formula: Show the active shape and the closed-form Ix and Iy expression in a formula box, so the answer is auditable against a textbook or Wolfram MathWorld entry.
• • Built-in area sanity check: Print the cross-section area in m^2 alongside Ix and Iy, so a typo in a single dimension is visible before the inertia number is fed into a beam equation.

The same form runs the closed-form lookup for all six shapes, so the only thing that changes is which dimension fields are read.

For a generic rectangle, triangle, or parallelogram that is not one of the six cross-sections here, the Area Calculator covers the same shape family in plain square units.

Three things move the Ix and Iy numbers more than anything else, plus the unit system. Knowing which lever to pull keeps a beam or shaft result defensible.

• • Centroidal only. If the bending axis is not the centroid, apply the parallel-axis theorem I_parallel = I_centroidal + A * d^2 by hand in the next step.
• • Single closed shape only. Composite sections, hollow shapes, and built-up beams are out of scope; for a hollow tube, subtract the inner moment of area from the outer one.

For most beam and shaft work, the centroidal Ix and Iy are exactly the inputs the next formula needs, and the polar J is the input the shaft torsion equation needs.

According to Wikipedia, the second moment of area is the integral of the squared distance from an axis over the area of the cross-section, and its SI unit is the metre to the fourth power (m^4).

When the cross-section is an ellipse and the next step is the area in plain square units, the Ellipse Area Calculator returns the same pi * a * b from the same two inputs.