SizingKitMechanical · Section properties
Moment of inertia calculator for area, not mass
This returns the area moment of inertia and the elastic section modulus of a cross-section — the two numbers that decide how far a member bends and how hard its outer fiber works — for a rectangle, a round bar, a tube, a box section or an I-shape, about the strong axis and the weak one at once. It is free, needs no signup, and it deliberately asks for no load, no span and no material: a section property belongs to the piece of metal and is the same whether that piece ends up a joist, a machine base or a bracket. If the formula you are holding has a mass in it, you want the mass moment of inertia instead, which is a different quantity in different units and is not what this page computes.
- 100% free
- No signup
- Both axes at once
- in⁴ and cm⁴
- Parallel axis theorem
The section
Depth is always the dimension the load runs along, so a bar lying flat and the same bar stood on edge are two different entries here rather than one entry and a setting.
A unit typed after a figure wins over this one.
The thin way of a joist standing up. Enters the answer once.
The tall way. Enters the answer as a cube, which is the whole point.
I, STRONG AXIS
47.63 in⁴
1,983 cm⁴
S, STRONG AXIS
13.14 in³
215.3 cm³
I, WEAK AXIS
2.039 in⁴
84.87 cm⁴
| Property | About the strong axis | About the weak axis | ||
|---|---|---|---|---|
| Area A | 10.88 in² | 7,016 mm² | 10.88 in² | 7,016 mm² |
| Second moment of area I | 47.63 in⁴ | 1,983 cm⁴ | 2.039 in⁴ | 84.87 cm⁴ |
| Section modulus S | 13.14 in³ | 215.3 cm³ | 2.719 in³ | 44.55 cm³ |
| Centroid to extreme fiber c | 3.625 in | 92.07 mm | 0.75 in | 19.05 mm |
| Radius of gyration r | 2.093 in | 53.16 mm | 0.433 in | 11 mm |
What it rests on: I = b·h³ / 12 about the strong axis and h·b³ / 12 about the weak one, with b across the load, h along it. S is I divided by 3.625 in, the distance from the centroid out to the furthest fiber.
Turned the other way this section is 23.4 times less stiff — 2.039 in⁴ against 47.63 in⁴ — for exactly the same weight of metal per foot. That ratio is the only thing orientation buys, and it costs nothing, which is why a joist is installed on edge and a length of angle used flat is nearly always a mistake somebody made in a hurry.
These are second moments of area, reported in length to the fourth power: in⁴, cm⁴, mm⁴. The other quantity that goes by the name moment of inertia is the mass moment, the resistance of a spinning part to being sped up, and it is measured in lb·in² or kg·m². Nothing on this page computes it. The units are the fastest way to tell which one a formula wants — if the expression has a mass in it, it is not this one, and rotating-drive work belongs with gear ratio rpm calculator instead.
The box and I-shape cases square off their corners. A formed tube has a radius inside and out and a rolled beam has fillets where the web meets the flange, so the figure above runs slightly over a real tube and slightly under a real rolled shape. For a listed HSS or W shape the published second moment is the one to use: this site does not reproduce the AISC shape tables, and interpolating a listed section out of the geometry above is not the same number.
Built up from parts
A plate welded to a flange, two bars with a spacer between them, a fabricated beam out of flat stock. Each row is a rectangle; the third column says how high its own center sits above whatever datum you are measuring from — the bottom of the assembly is the easiest one.
| Part | Width b | Depth h | Its center, above the datum |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 |
COMPOSITE I
47.85 in⁴
1,992 cm⁴
NEUTRAL AXIS, ABOVE DATUM
0.3565 in
9.056 mm
SMALLER SECTION MODULUS
13 in³
the bottom face is furthest from the axis
| Total area of 2 parts | 13.13 in² | 8,468 mm² |
|---|---|---|
| Sum of the parts' own second moments | 47.66 in⁴ | 99.6% of the total |
| Transfer term, Σ A·d² | 0.1882 in⁴ | 0.393% of the total |
| Section modulus to the top face | 13.4 in³ | c = 3.571 in |
| Section modulus to the bottom face | 13 in³ | c = 3.679 in |
| Radius of gyration r | 1.909 in | 48.5 mm |
The transfer term carries 0.393% of this section, and the parts' own second moments carry the rest. That split is the theorem's entire practical content: a part contributes A·d², so distance is squared and thickness is not, and a thin piece welded far from the axis does more than a heavy one bolted near it. Because the parts sit at different heights the neutral axis has moved off center, which leaves two section moduli rather than one — 13.4 in³ to the top face and 13 in³ to the bottom. Bending stress is read against the smaller of the two, since that face is the one furthest from the axis and therefore the most stressed.
The five sections, as formulas
What the panel above evaluates, written out. Every entry is in length to the fourth power for I and the third for S, so a sheet printed from here works in inches or in millimeters as long as one set of units goes in throughout.
| Section | Area | I, strong axis | I, weak axis |
|---|---|---|---|
| Solid rectangle b across the load, h along it | b·h | b·h³ / 12 | h·b³ / 12 |
| Solid round d the diameter | π·d² / 4 | π·d⁴ / 64 | π·d⁴ / 64 |
| Round tube D outside, d bore | π·(D² − d²) / 4 | π·(D⁴ − d⁴) / 64 | π·(D⁴ − d⁴) / 64 |
| Box section b across, h along, t the wall | b·h − (b−2t)·(h−2t) | [ b·h³ − (b−2t)·(h−2t)³ ] / 12 | [ h·b³ − (h−2t)·(b−2t)³ ] / 12 |
| I-shape b flange width, h depth, tf flange, tw web | b·h − (b−tw)·(h−2tf) | [ b·h³ − (b−tw)·(h−2tf)³ ] / 12 | [ 2·tf·b³ + (h−2tf)·tw³ ] / 12 |
S = I ÷ c in every row, with c half the depth for the strong axis and half the width for the weak one, because all five sections are symmetric about both. An unsymmetric section — a channel on its side, an angle, anything built up — has a different c at each face and therefore two section moduli, which is what the built-up panel reports. The standard geometry, as tabulated in the AISC Steel Construction Manual and in Roark’s Formulas for Stress and Strain.
Everything here describes a section rather than a member. Handing I and S to beam deflection calculator is what turns them into a deflection and a stress, and that takes a span, a load case and an elastic modulus that this page never asks for.
How to get a section property out of a drawing
Three decisions: which shape, which way up, and whether it is one piece or several.
Say which way up the section sits
Pick the shape, then read the two dimension labels carefully: width is measured across the load and depth along it. That is not a formality. A 1.5 by 7.25 in bar entered as 1.5 wide and 7.25 deep returns 47.63 in⁴, and the same bar entered the other way round returns 2.039 in⁴, because depth is cubed and width is not. If the member is a joist it is standing up; if it is a shelf it is lying down; the entry has to match the installation rather than the way the size is spoken.
Read both axis columns, not just the first
The panel returns the strong axis and the weak axis side by side because a section only resists bending in the plane it is loaded in. A joist that carries floor load beautifully has almost no stiffness sideways, which is why it needs blocking; a round bar has the same figure in every direction and needs none. Where the two columns are far apart the orientation is doing most of the work, and where they are equal the shape has no preferred direction at all.
Build up anything that is not one of the five
A channel with a plate across the toes, two bars with a spacer, a fabricated beam out of flat stock: enter each piece as a rectangle with the height of its own center above a datum you choose, and the lower panel finds the combined neutral axis and applies the parallel axis theorem about it. It reports the transfer term separately from the parts' own second moments, so the split between the two is visible — and it reports two section moduli, because a built-up section is rarely symmetric and the smaller one is the figure a stress calculation wants.
Technical specifications
| Sections | Solid rectangle, solid round, round tube, box section and I-shape, each returned about both principal axes. The I-shape's weak axis is assembled from three rectangles sharing one centroidal axis, because an I-shape turned on its side is not an I-shape |
|---|---|
| Figures returned | Area, second moment of area I, elastic section modulus S, distance to the extreme fiber c, and radius of gyration r — five per axis, ten per section |
| Units | Inches or millimeters in, and both systems out at once: in² and mm² for area, in⁴ and cm⁴ for I, in³ and cm³ for S. European section tables print cm⁴ because a rolled beam in mm⁴ runs to eight digits |
| The orientation penalty | A 1.5 × 7.25 in section gives 47.63 in⁴ on edge and 2.039 in⁴ flat — a factor of 23.4, which is the square of the 4.83 aspect ratio, at identical weight per foot |
| What a tube buys | A 2 in outside diameter tube with a 0.125 in wall keeps 41.4% of the solid bar's second moment while carrying 23.4% of its area, so it is 1.77 times as stiff per pound of steel |
| What an I-shape buys | A 4 × 8 in I-shape with 1/4 in flanges and web is 3.875 in² in area and 38.83 in⁴ about the strong axis. A solid rectangle of the same area, 1.5 in wide, reaches 2.16 in⁴ — 18 times less, for the same steel |
| Built-up sections | Four rectangular parts, each placed by the height of its own center above a datum. Returns the combined neutral axis, Σ(Ic + A·d²) about it, both section moduli, and the transfer term separately from the parts' own second moments |
| Worked built-up example | A 6 × 3/8 in plate laid on top of a 1.5 × 7.25 in bar adds 21% to the area and takes I from 47.63 to 74.76 in⁴, a 57% gain. The plate's own second moment is 0.0264 in⁴ of that; the rest is the A·d² transfer |
Frequently asked questions
Is this the same moment of inertia as the one in the flywheel formula?
No, and the units are the quickest way to tell them apart. This page returns the second moment of area, in inches or millimeters to the fourth power, which describes how a cross-section resists bending. The flywheel quantity is the mass moment of inertia, in lb·in² or kg·m², which describes how a lump of material resists being spun up faster. They share a symbol and a name and nothing else: one is a property of a shape drawn on paper, the other is a property of an object with a weight. If the formula in front of you has a mass in it — the ½mr² of a disc, the mr² of a point — it wants the second one, and nothing here computes it.
Why does swapping the two dimensions change the answer so much?
Because depth enters as a cube and width enters once, so the ratio between the two orientations is the square of the aspect ratio. A 1.5 by 7.25 in section is 4.83 times deeper than it is wide, and 4.83 squared is 23.4 — which is exactly the factor between 47.63 in⁴ on edge and 2.039 in⁴ flat, for identical metal and identical weight per foot. Nothing else in section work pays that well for free, which is why every joist, every rafter and every piece of angle used as a lintel is installed the tall way, and why a plank walked on flat feels like a different piece of timber from the same plank on edge.
Can I use the I-shape entry for a rolled W or S beam?
It will give you a close figure and not the published one, and for anything the beam is being sized for you want the published one. The geometry here is the enclosing rectangle less two rectangular voids, which is exact for a plate girder welded up with square corners. A rolled shape has a fillet radius where each flange meets the web, and that extra metal sits close to the neutral axis where it adds area faster than it adds second moment — so the real section is a little stiffer and considerably heavier than the square-cornered arithmetic suggests. Section properties for listed W, S, C, HSS and angle shapes come out of the AISC Steel Construction Manual shape tables, which this site does not reproduce.
If I already have I, what is the section modulus for?
Different question, different answer: I governs how far the member moves and S governs how hard the outer fiber works. Deflection is proportional to 1 over I, so doubling the second moment halves the sag. Bending stress is the moment divided by S, so doubling S halves the stress. The two are related by S = I divided by c, the distance from the neutral axis out to the furthest fiber, and that division is what makes them behave differently as a section gets deeper: going from a 6 in deep section to a 12 in deep one of the same width multiplies I by eight and S by only four.
My section is a channel, an angle or a tee — what do I do?
Decompose it into rectangles in the built-up panel, with one caveat that matters for the angle. A channel and a tee are symmetric about the axis of the web, so breaking them into rectangles and applying the parallel axis theorem gives the right answer for bending in that plane. An equal-leg angle is not symmetric about either the horizontal or the vertical: its stiffest and least stiff axes are rotated 45 degrees from the legs, so an angle loaded straight down bends and twists at once and moves sideways as well as downward. The rectangles you draw will return the second moment about the axis you drew, which is a real number about a real axis and is not the principal axis the piece actually wants to bend about.
Does drilling a hole in a beam wreck it?
It depends almost entirely on where the hole is, and the parallel axis theorem is why. A piece of material contributes its own second moment plus its area times the square of its distance from the neutral axis, so metal near the axis is contributing almost nothing to bending stiffness and metal out at the flange is contributing nearly all of it. A hole punched through the web at mid-depth costs very little; the same hole through a flange costs a great deal, because the area removed is multiplied by a large squared distance. That is also the argument for the I-shape itself: it is a rectangle with the useless middle taken out.
Which figure do I use for a column rather than a beam?
The radius of gyration, and it is the smaller of the two the panel prints rather than the larger. A column buckles about whichever axis is weakest, so the slenderness that governs is the unbraced length divided by the smaller r — and for a section with two very different axes, that is why a deep narrow beam used upright as a post is a poor idea unless something braces it sideways. This page returns r as geometry and stops there. Turning a slenderness into a capacity takes a buckling curve and a material strength out of the governing design standard, and it is a different calculation from anything on this page.
About the second moment of area, and the two things it is confused with
The second moment of area is an integral of area against the square of distance from an axis, and everything that is unintuitive about section design comes out of that square. Material sitting on the neutral axis contributes essentially nothing to bending stiffness; material out at the extreme fiber contributes in proportion to its distance squared. That is the whole argument for the I-shape, which is a rectangle with the unproductive middle removed, and for the tube, which is a bar with the same middle removed in the round. It is also why depth appears as a cube in every solid formula on this page while width appears once — depth adds area and moves that area away from the axis at the same time, so it pays twice.
The name is shared by a completely different quantity, and the confusion is the single most common reason someone lands on a page like this one and leaves with the wrong number. The area moment has units of length to the fourth and describes a shape; the mass moment has units of mass times length squared and describes an object, telling you what torque it takes to spin that object up rather than what load it takes to bend it. A machine designer sizing the shaft of a drive needs both at different moments — the area moment when the shaft is a beam between two bearings, the mass moment when the same shaft is a rotating inertia to be accelerated, which is where the gear ratio and RPM page takes over. Two rules of thumb keep them apart: if the expression contains a density or a weight it is not the one on this page, and if the answer comes out in in⁴ it is.
What most section calculators get wrong is not the arithmetic but the silence about which axis they answered. A rectangle has two second moments that differ by the square of its aspect ratio, and a page that prints one number without saying which orientation produced it is guessing on the reader's behalf. This one prints both columns every time, and for a built-up section it prints two section moduli as well, because welding a plate to one face moves the neutral axis off center and the two faces stop being the same distance from it. The figures then go somewhere: a second moment becomes a deflection only when it meets a span and a load case on the beam deflection page, and it becomes a weight per foot only when it meets a density on the metal weight page. Nothing here is a stamped design; it is the geometry that everything downstream depends on, and the engineer of record still owns the member, the connection and the load path.
Where the dimensions you type end up
Every number on this page is worked out by JavaScript running in the tab you are reading it in. Nothing you type — loads, lengths, nameplate ratings, the rates your utility charges you — is uploaded, logged or kept, which is also why the calculators carry on working in a mechanical room with no signal.
Shape dimensions and built-up part tables live in this tab and nowhere else — no drawing you are working from is transmitted, and a printed section sheet is generated locally rather than fetched, so it leaves no record of the job it was for.