Parallel Axis Theorem Calculator

Combine centroidal moment of inertia, area, and offset distance to find the composite second moment of area about a new reference axis.

Configuration

Choose what you want to solve for and the reporting units for the moment of inertia. The inputs below will adjust automatically so that the solved quantity is hidden and treated as unknown.

Section Properties

Enter the centroidal inertia, area, and offset distance for the component you are shifting. All inputs support both SI and imperial units and are converted consistently before the parallel axis theorem is applied.

Results Summary

The main result is shown below, with quick stats for key derived quantities like equivalent radius of gyration and reported values in multiple unit systems.

Practical Guide

Parallel Axis Theorem Calculator: From Centroid Data to Real-World Sections

Learn how to use the Parallel Axis Theorem Calculator to move second moments of area from centroidal axes to any parallel reference axis. See how to combine shapes, subtract holes, interpret units, and sanity-check your results against real beams and machine components.

10–14 min read Updated 2025

Quick Start: Using the Parallel Axis Theorem Calculator Safely

The parallel axis theorem relates the second moment of area about a centroidal axis to a parallel axis some distance away. The core equation the calculator uses is:

\[ I_p = I_c + A d^2 \]

where \(I_p\) is the second moment of area about the chosen reference axis, \(I_c\) is about the centroidal axis, \(A\) is area, and \(d\) is the perpendicular distance between the two parallel axes.

  1. 1 Decide which quantity you want the calculator to solve for: transformed moment of inertia \(I_p\), the required distance \(d\)missing centroidal value \(I_c\).
  2. 2 Choose the section type. For basic shapes (rectangle, circle, hollow section), you can often let the calculator compute \(A\) and \(I_c\) from dimensions. For built-up/custom sections, you may enter \(A\) and \(I_c\) directly.
  3. 3 Set a clear coordinate system and reference axis. Decide whether you are shifting to a beam top/bottom fiber, a connection plate, a shaft centerline, or another design axis. The distance \(d\) must be measured perpendicular to the axes you are using.
  4. 4 Enter all dimensions and areas in consistent units (for example, mm and mm², or in and in²). If you mix metric and imperial units, the numbers will be off by orders of magnitude.
  5. 5 For composite sections, add one component at a time. For each component, use its centroidal \(I_c\), area \(A\), and distance \(d\) to the global axis, then sum or subtract contributions according to whether it is material or a void.
  6. 6 Review the Quick Stats and Calculation Steps under the calculator. Check that the resulting \(I_p\) is in the right ballpark compared with catalog values or similar sections.
  7. 7 Use the result in your deflection, bending stress, and stability checks (for example, \( \sigma = M y / I \) or \( \delta \propto L^3 / (E I) \)). Always verify code-specific requirements for which axis and section properties to use.

Tip: Sketch the section and axes before you type anything. Most parallel axis mistakes come from confusing which centroid-to-axis distance is actually \(d\).

Warning: The parallel axis theorem is valid only for parallel axes. It does not replace a full coordinate transformation when axes are rotated.

Choosing Your Method: Single Shape, Composite, or Tabulated Values

There are several ways engineers use the parallel axis theorem in practice. The calculator supports all of them; you just choose the mode that matches how your problem is defined.

Method A — Shift a Single Basic Shape

Ideal when you know the centroidal second moment of area \(I_c\) from a textbook or formula and just need to move it to a different reference axis.

  • Fast and clean for rectangles, circles, thin plates, and standard bars.
  • Perfect for moving a centroidal axis to a top or bottom fiber of a beam.
  • Matches most textbook examples exactly.
  • Only handles one shape at a time.
  • Does not directly model stiffeners, cutouts, or built-up members.
Basic form: \( I_p = I_c + A d^2 \)

Method B — Composite Sections (Add and Subtract Pieces)

Use this when your section is built from several rectangles, plates, or flanges, or when you have cutouts and holes.

  • Works for I-beams, channels, T sections, box girders, and plates with holes.
  • Directly aligns with many design code examples for built-up members.
  • Lets you see which component contributes most to stiffness.
  • Requires careful bookkeeping of areas, centroids, and signs (add vs subtract).
  • Easy to mislabel distances when there are multiple reference axes.
Composite: \( I_p = \sum\limits_i \bigl(I_{c,i} + A_i d_i^2\bigr) \)

Method C — Start from Tabulated Section Properties

Many steel, aluminum, and FRP shapes come with published section properties about standard axes. Use the calculator to adjust for non-standard axes.

  • Leverages accurate catalog data for hot-rolled and cold-formed shapes.
  • Reduces derivation time when code tables already define \(I_c\).
  • Useful for checking custom sections against a nearby catalog size.
  • Only as good as the tabulated data and axis definitions.
  • You still need to keep units and signs consistent when shifting axes.
Catalog-based: use \(I_c\) from the table → apply \(I_p = I_c + A d^2\) to your axis.

What Moves the Number: Key Drivers in the Parallel Axis Theorem

The parallel axis theorem looks simple, but in design the numbers can change by an order of magnitude with small modelling decisions. These are the main levers that change \(I_p\).

Area \(A\)

Larger area means a stronger contribution to \(I_p\). When you add a plate or stiffener, you are adding both its centroidal \(I_c\) and the \(A d^2\) term, so even thin plates can matter if they are far from the axis.

Distance \(d\)

The distance is squared in \(A d^2\), so moving material further from the axis is extremely powerful. Doubling \(d\) increases the parallel-axis contribution by a factor of four.

Reference axis location

The same shape has different moments of inertia about different axes. Shifting from the centroid to the edge of a beam, or from one flange to another, can change the value dramatically.

Sign convention

For added material, areas are positive; for holes or voids, areas are modelled as negative. The parallel axis theorem still applies, but a void reduces the net \(I_p\).

Axis orientation (x, y, and polar)

In 2D, you usually work with \(I_x\) and \(I_y\). For torsion problems, the polar moment of inertia \(J\) can be found from \(J = I_x + I_y\) about the same point, often combined with a parallel-axis shift.

Unit system and scaling

Changing from mm to m multiplies dimensions by 1000; second moments of area scale with length to the fourth power \((L^4)\). Make sure all inputs live in the same unit system before using the result in formulas.

Worked Examples: Parallel Axis Theorem in Action

Example 1 — Shifting a Rectangular Plate to Its Bottom Edge

  • Rectangular plate: width \(b = 100\ \text{mm}\), height \(h = 20\ \text{mm}\).
  • Material is uniform; thickness is out of plane and not needed for this 2D section property.
  • We know centroidal second moment of area about the horizontal axis through the centroid: \[ I_{c} = \frac{b h^3}{12}. \]
  • We want the second moment of area about the bottom edge, which is parallel to the centroidal axis.
1
Compute \(I_c\) about centroid: \[ I_c = \frac{b h^3}{12} = \frac{100 \times 20^3}{12} = \frac{100 \times 8000}{12} \approx 6.67 \times 10^4\ \text{mm}^4. \]
2
Compute area and distance: \[ A = b h = 100 \times 20 = 2000\ \text{mm}^2. \] The centroid is at mid-depth, so the distance from the centroidal axis to the bottom edge is \[ d = \frac{h}{2} = 10\ \text{mm}. \]
3
Apply parallel axis theorem: \[ I_p = I_c + A d^2 = 6.67 \times 10^4 + 2000 \times 10^2 = 6.67 \times 10^4 + 2.00 \times 10^6 \approx 2.07 \times 10^6\ \text{mm}^4. \] The bottom edge axis is much stiffer because of the \(A d^2\) term.
4
Use the calculator: In the calculator, select a rectangular section, enter \(b\), \(h\), and pick “Solve for \(I_p\)” with the bottom edge as the reference axis. The steps displayed will mirror this derivation.

Example 2 — Composite T-Section About Its Centroidal Axis

  • Flange: \(b_f = 200\ \text{mm}\), thickness \(t_f = 20\ \text{mm}\).
  • Web: thickness \(t_w = 20\ \text{mm}\), height \(h_w = 200\ \text{mm}\) measured below the flange.
  • We want the second moment of area about a horizontal axis through the composite centroid (often used for beam bending about the strong axis).
1
Compute areas and centroids: \[ A_f = b_f t_f = 200 \times 20 = 4000\ \text{mm}^2, \] \[ A_w = t_w h_w = 20 \times 200 = 4000\ \text{mm}^2. \] Choose the bottom of the web as \(y = 0\). Then \[ y_f = h_w + \frac{t_f}{2} = 200 + 10 = 210\ \text{mm}, \] \[ y_w = \frac{h_w}{2} = 100\ \text{mm}. \] The composite centroidal location is \[ \bar{y} = \frac{A_f y_f + A_w y_w}{A_f + A_w}. \]
2
Find distance of each component from the composite centroid: After computing \(\bar{y}\), determine \[ d_f = |y_f – \bar{y}|,\quad d_w = |y_w – \bar{y}|. \] These distances are the \(d\) values used in the parallel axis theorem for each rectangle.
3
Apply the theorem to each piece: For rectangles, the centroidal moment of inertia about their own horizontal axes is \[ I_{c,f} = \frac{b_f t_f^3}{12},\quad I_{c,w} = \frac{t_w h_w^3}{12}. \] Then \[ I_{\text{T}} = (I_{c,f} + A_f d_f^2) + (I_{c,w} + A_w d_w^2). \]
4
Check against the calculator: In composite mode, the calculator lets you enter each component’s dimensions, automatically computes areas, centroids, \(I_c\), and distances to the global axis, and then sums them. Use the step-by-step output to confirm your manual spreadsheet or hand calculation.
Key relationship used in both examples
\[ I_p = \sum_i \bigl(I_{c,i} + A_i d_i^2\bigr) \]

Common Layouts & Variations

The parallel axis theorem appears in many different layouts: thin plates, stiffened beams, shafts with keyways, and perforated panels. The calculator helps you model these consistently.

ConfigurationTypical Use CaseParallel Axis SetupCommon Pitfalls
Rectangular section about edgeSimple beams, plates supported on one sideUse rectangle formulas for \(I_c\); set \(d = h/2\) from centroid to edge axisForgetting that \(I_c\) changes if you rotate the rectangle (swap \(b\) and \(h\))
Built-up I or T sectionWelded girders, stiffened platesModel flange and web as separate rectangles; compute composite centroid, then apply parallel axis theoremMixing reference for \(y\); each component must use the same global axis when evaluating \(d\)
Section with circular holeLightweight beams, access openings, piping supportsModel the hole as negative area and subtract \(I_c + A d^2\) of the circular voidForgetting to subtract both the hole area and its parallel-axis contribution
Offset plate or stiffenerAdding stiffness far from neutral axisUse plate area and centroid, then apply large \(d\) in \(A d^2\)Entering dimensions in mixed units (e.g., mm and in) leading to unrealistic stiffness
Polar moment about a non-centroidal pointShaft torsion, bolted circular patternsUse \(J_O = I_x + I_y\) about point \(O\), where each \(I\) is adjusted via the parallel axis theoremApplying polar formulas for solid circles directly when the axis is not through the center
  • Draw axes and centroids clearly on your sketch before computing \(d\).
  • Keep one consistent coordinate system for all components.
  • Check the final \(I_p\) against a similar catalog section when possible.
  • Use symmetrical sections as quick sanity checks; centroids should line up with symmetry lines.
  • Remember that second moments of area are geometric properties, independent of material \(E\).
  • For rotated or skew axes, use full transformation equations, not the parallel axis theorem alone.

Specs, Logistics & Sanity Checks

While the parallel axis theorem is a mathematical identity, the way you apply it in design is influenced by code requirements, drawing conventions, and how models are shared across teams.

Specifying Section Properties

In reports and design summaries, clearly state:

  • Which axis each reported \(I\) value is about (e.g., strong axis, weak axis, top-fiber axis).
  • The unit system, e.g., mm\(^4\), cm\(^4\), in\(^4\).
  • Whether section properties include or exclude holes, cutouts, or attachments.

Use the calculator to generate both centroidal and shifted moments so reviewers can cross-check your assumptions.

Modelling and Coordination

In FEA models and BIM workflows, the axis definitions may differ from hand calcs. Align on:

  • Global versus local axes for beams and shells.
  • Which axis is used for bending versus torsion checks.
  • How composite properties are constructed when stiffeners or overlays are added.

The calculator is a quick way to verify that model-reported section properties make sense for your simplified hand models.

Sanity Checks Before You Trust the Number

  • Does \(I_p\) increase when you move area farther from the axis? If it decreases, a sign or distance is wrong.
  • For symmetric sections, is the centroid where you expect it (e.g., mid-depth or mid-width)?
  • Do your values fall within the range of similar catalog sections?
  • For very large or very small numbers, check that you did not accidentally mix mm and m.

A quick rule of thumb: if you double a dimension everywhere, second moments of area should increase by a factor of \(2^4 = 16\). Use this as a rough scaling check.

Finally, remember that the parallel axis theorem is only part of the story. When you plug \(I_p\) into beam, column, or shaft formulas, verify that your load cases, boundary conditions, and material models align with how the geometric property was defined.

Frequently Asked Questions

What is the parallel axis theorem in simple terms?
The parallel axis theorem says that the second moment of area about any axis parallel to a centroidal axis is equal to the centroidal second moment of area plus the area multiplied by the square of the distance between the axes: \(I_p = I_c + A d^2\). It lets you move section properties from a convenient axis (usually through the centroid) to the actual design axis.
When should I use the Parallel Axis Theorem Calculator?
Use the calculator whenever you need a moment of inertia about an axis that is not coincident with the centroidal axis. Typical cases include computing \(I\) about a top or bottom fiber, evaluating stiffness of built-up sections, or adjusting catalog section properties to a different reference axis.
Can I use the parallel axis theorem for composite sections and holes?
Yes. For composite sections, apply the theorem to each component and sum their contributions. For holes or voids, treat the area as negative and subtract \(I_c + A d^2\) for the hole. The calculator can reflect this by letting you define components as either material or cutout and combining their effects.
Does the parallel axis theorem work for polar moment of inertia?
It can. The polar moment of inertia about a point is \(J = I_x + I_y\) about that same point. If you know centroidal values \(I_{x,c}\) and \(I_{y,c}\), you can use the parallel axis theorem to shift both to the new point, then add them to get \(J\). The calculator can help by computing the shifted \(I_x\) and \(I_y\) first.
What are the most common mistakes when applying the parallel axis theorem?
Common mistakes include using the wrong distance \(d\) (measuring along the axis instead of perpendicular to it), mixing units (mm with m or in), applying the theorem to non-parallel axes, and forgetting to subtract the contribution of holes or cutouts. The calculator mitigates many of these by enforcing consistent inputs and showing intermediate steps.
Is the parallel axis theorem enough for rotated or inclined axes?
No. The theorem only relates parallel axes. For rotated or inclined axes, you need full Mohr’s circle or coordinate transformation equations for second moments of area. You can still use the parallel axis theorem as part of the process, but you must also account for axis rotation.
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