Centroids and Moments of Inertia - Cheat Sheet
Your last-minute revision companion for Engineering Mechanics: centroids, moments of inertia, parallel-axis theorem, and radius of gyration. Every formula, definition, and board-exam pitfall condensed into one sheet.
Centroid of Composite Areas
Formulas
x̄ = (ΣAᵢxᵢ) / ΣAᵢ; ȳ = (ΣAᵢyᵢ) / ΣAᵢ
Meaning
x̄, ȳ = centroid coordinates; Aᵢ = area of part i; xᵢ, yᵢ = coordinates of part i centroid
Watch out
HOLES ARE NEGATIVE AREAS — subtract both Ahole and its (xhole, yhole) contributions. Sign error here kills half the section.
When to use
Always — to find geometric center of any composite (built-up) section
Rectangle: (x̄, ȳ) = (b/2, h/2) from corner
Meaning
b = width, h = height
Watch out
This is from the corner/origin; if origin is elsewhere, recalculate.
When to use
Rectangular parts in composite sections
Triangle: ȳ = h/3 from base; x̄ depends on orientation
Meaning
h = height measured perpendicular to base
Watch out
h/3, NOT h/2. Common error: confusing with rectangle centroid at h/2.
When to use
Triangular parts in composite L-, T-, or trapezoidal sections
Semicircle: distance from diameter = 4r/(3π)
Meaning
r = radius
Watch out
Memorize 4r/(3π) ≈ 0.424r. NOT r/2.
When to use
Semicircular flanges or cut-outs (e.g., architectural sections)
Quarter-circle: distance from each straight edge = 4r/(3π)
Meaning
r = radius
Watch out
Same as semicircle: 4r/(3π) in BOTH x and y directions.
When to use
Quarter-circle fillets or sections
Common values
Value
(b/2, h/2)
Symbol
x̄rect, ȳrect
Quantity
Centroid of rectangle from corner
Value
h/3 perpendicular to base
Symbol
ȳtri
Quantity
Centroid of triangle from base
Value
4r / (3π) ≈ 0.424r
Symbol
ȳsemi
Quantity
Centroid of semicircle from diameter
Important facts
- Centroid is ALWAYS on every axis of symmetry — use this to reduce calculation.
- Holes (cut-outs) are treated as NEGATIVE areas: subtract both A and coordinates.
- For symmetric sections (I-beam, channel, angle), one coordinate is zero or obvious by symmetry.
- Origin location matters: always state the reference point (corner, base, etc.).
- Centroid does NOT have to be inside the material (e.g., hollow tube or angle section).
Key definitions
Centroid
Example
A 100×100 mm square has centroid at (50, 50) mm from one corner.
The geometric center (balance point) of an area, found as the area-weighted average of coordinates.
Composite Area
Example
T-section = rectangle (flange) + rectangle (web); L-section = two rectangles.
A complex shape made of simple standard shapes (rectangles, triangles, circles) added or subtracted.
Axis of Symmetry
Example
A square has 4 axes of symmetry; the centroid is at the intersection (center).
A line about which a shape is mirror-symmetric; the centroid always lies on every axis of symmetry.
Diagrams to know
- T-section with centroid marked (flange top, web bottom)
- L-section with divided areas (two rectangles, centroid shown)
- Composite rectangle with circular hole — centroid shift illustrated
- Trapezoid divided into simple shapes (rectangle + triangles)
Reactions or equations
x̄ = Σ(Aᵢ × xᵢ) / ΣAᵢ (same for ȳ with yᵢ)
Note
This is the definition of centroid. Every composite section problem starts here.
Conditions
All areas in same unit system; coordinates measured from same origin
Acomposite = ΣAᵢ − Aholes
Note
Total area is sum of parts minus holes. Used in deflection, stress formulas later.
Conditions
Standard addition/subtraction of areas
Moment of Inertia (Second Moment of Area)
Formulas
Ix = ∫y² dA; Iy = ∫x² dA
Meaning
Ix, Iy = moments of inertia about x and y axes; y, x = perpendicular distances from axis
Watch out
This is an integral (sum of area elements times squared distance). Don't confuse with first moment (centroid).
When to use
Definition of moment of inertia. Used to derive standard formulas below.
Rectangle (about centroidal axis): I = bh³/12
Meaning
b = width parallel to axis, h = depth perpendicular to axis
Watch out
bh³/12 is about CENTROID. For moment about the base: Ibase = bh³/3. Check your reference point!
When to use
Any rectangular part of a composite section (transfer with parallel-axis theorem if off-center)
Rectangle (about base axis): I = bh³/3
Meaning
b = width, h = height from base
Watch out
Ibase = Icentroid + A(h/2)² = bh³/12 + (bh)(h/2)² = bh³/12 + bh³/4 = bh³/3 ✓
When to use
Rarely used directly in composite sections (use parallel-axis theorem instead)
Triangle (about centroidal axis): I = bh³/36
Meaning
b = base width, h = height perpendicular to base
Watch out
bh³/36 is about CENTROID (at h/3 from base). Not bh³/12.
When to use
Triangular parts (e.g., trapezoidal section divided into rectangle + triangle)
Triangle (about base axis): I = bh³/12
Meaning
b = base width, h = height
Watch out
Ibase = Icentroid + A d² = bh³/36 + (bh/2)(h/3)² = bh³/36 + bh³/18 = bh³/12 ✓
When to use
Less common (use parallel-axis to shift to centroid if needed)
Circle (about centroid): I = πd⁴/64 = πr⁴/4
Meaning
d = diameter, r = radius
Watch out
πd⁴/64 uses diameter. If given radius: πr⁴/4. Both equal; don't mix.
When to use
Circular sections or holes; same Ix = Iy by symmetry
Hollow Rectangle: I = (b₁h₁³ − b₂h₂³)/12
Meaning
b₁, h₁ = outer width/height; b₂, h₂ = inner (hole) width/height; centroid shared
Watch out
Only works if centroids coincide. If off-center: calculate each part separately with parallel-axis.
When to use
When inner and outer rectangles are concentric (same centroid)
Common values
Value
bh³/12
Symbol
Irect,c
Quantity
Rectangle centroidal moment
Value
bh³/3
Symbol
Irect,base
Quantity
Rectangle base moment
Value
bh³/36
Symbol
Itri,c
Quantity
Triangle centroidal moment
Value
πd⁴/64 or πr⁴/4
Symbol
Icircle
Quantity
Circle moment
Important facts
- Moment of inertia ALWAYS increases with distance from an axis (proportional to distance²).
- For a given area, SPREAD the material as far as possible from the axis to maximize I (e.g., I-beam flanges far from web).
- Moment of inertia is NEVER negative; it's always positive and increases with d².
- Square sections have Ix = Iy; rectangular sections have Iwide > Inarrow (compare to weak axis).
- Hollow sections (tubes, boxes) have high I-to-weight ratio — preferred in design.
Key definitions
Moment of Inertia (Second Moment of Area)
Example
A tall, narrow rectangle has large I about its weak axis; a square has equal I in both directions.
A measure of how an area is distributed about an axis, quantifying resistance to bending and buckling. Units: mm⁴ or m⁴.
Centroidal Axis
Example
For a rectangle, the centroidal x-axis is at y = h/2 from the base.
An axis passing through the centroid of the shape.
Principal Axes
Example
For a rectangle or T-section, principal axes are usually horizontal and vertical.
Orthogonal axes about which Imax and Imin occur; for symmetric shapes, they coincide with axes of symmetry.
Diagrams to know
- Rectangle with bh³/12 formula applied to centroidal axis
- Triangle with centroid at h/3, moment arms shown
- Comparison: same-area rectangle vs. tall narrow rectangle (illustrates I effect)
- Hollow rectangle with inner/outer boundaries labeled
Reactions or equations
I = ∫y² dA definition; standard formulas derived by integration
Note
This integral is the foundation. AISC 360 and design codes rely on tabulated I values.
Conditions
y measured perpendicular from the axis of interest
Ibase = Icentroid + A(d)² where d = distance from centroid to base
Note
This is the parallel-axis theorem. Rearranged, it proves Ibase > Icentroid always.
Conditions
Axis parallel to centroidal axis
Parallel-Axis Theorem (THE WORKHORSE)
Formulas
I = Ī + Ad²
Meaning
I = moment of inertia about any axis; Ī = moment about centroidal axis; A = area; d = distance from centroid to new axis
Watch out
BIGGEST BOARD MISTAKE: Using only Ī (centroidal) and forgetting Ad². This UNDERESTIMATES I by huge amounts for off-axis parts. The Ad² term is 50%+ of total I for many sections.
When to use
ALWAYS for composite sections. Every single part needs this transfer to the composite centroid before summing.
I_composite = Σ(Ī_i + A_i d_i²) − (Ī_holes + A_holes d_holes²)
Meaning
Sum transferred inertias of all parts; subtract holes using NEGATIVE area and NEGATIVE terms
Watch out
HOLES are negative. If you forget to subtract a hole's Ī AND its Ad² term, you've massively overstated I.
When to use
Step-by-step procedure for T-sections, I-beams, L-sections, any built-up shape
Common values
Value
I_base / I_centroid = 3 (for rectangles)
Symbol
Ratio
Quantity
Multiplier for base moment vs centroidal
Value
Depends on geometry; flange ≈ h/3 to h/2 from centroid
Symbol
d
Quantity
Typical d value (T-section example)
Important facts
- The parallel-axis theorem is the #1 tested skill on the PRC exam for composite sections.
- You MUST locate the composite centroid FIRST before applying the theorem.
- Distance d is ALWAYS measured from the PART'S centroid to the COMPOSITE centroid — get the reference right.
- The Ad² term grows with d². For sections with parts far from the centroid, Ad² often dominates Ī.
- Forgetting this theorem on an exam results in dramatically incorrect answers (off by 2×, 3×, or more).
Key definitions
Parallel-Axis Theorem
Example
A rectangle's moment about the base is 5 times larger than about its centroid, due to the Ad² term.
A method to transfer moment of inertia from a centroidal axis to any parallel axis using I = Ī + Ad².
Composite Section Procedure
Example
T-section: divide into top flange and stem (web). Find composite ȳ. Transfer each part's I to ȳ line. Sum.
Step 1: Divide into simple parts. Step 2: Find overall centroid (x̄, ȳ). Step 3: For each part, calculate Ī and d (distance from part centroid to composite centroid). Step 4: Apply I = Ī + Ad² and sum.
Diagrams to know
- T-section with flange and web separated, showing d (distance from each part centroid to composite ȳ)
- Rectangle with centroidal axis vs. base axis, Ad² term illustrated
- L-section with two rectangles, d vectors shown for each, composite centroid marked
- I-beam with flanges and web, distance from flange centroid to I-beam centroid labeled
Reactions or equations
I_axis = I_centroid + A × d²
Note
Proven by integration; no exceptions. Always holds. Always use it for composite sections.
Conditions
d = perpendicular distance between the two parallel axes
d = |distance from part centroid to composite centroid|
Note
If you shift origins, recalculate d. Common error: inconsistent coordinate systems.
Conditions
Both centroid locations must be measured from the SAME origin
Polar Moment & Radius of Gyration
Formulas
J = Ix + Iy
Meaning
J = polar moment of inertia; Ix, Iy = moments about perpendicular centroidal axes
Watch out
J ≠ I about any single axis. J is the SUM of two perpendicular moments. For circles, Ix = Iy, so J = 2I.
When to use
Torsion problems, circular/symmetric sections. When you need resistance to twisting or overall radius of gyration.
Circle: J = πd⁴/32 = πr⁴/2
Meaning
d = diameter, r = radius; uses πd⁴/32 or πr⁴/2 (equivalent)
Watch out
J (polar) is TWICE the I (bending) for a circle. Don't confuse: I = πd⁴/64, but J = πd⁴/32.
When to use
Solid circular shafts, circular holes, polar moment of circular sections
Radius of Gyration: r = √(I/A)
Meaning
r = radius of gyration; I = moment of inertia; A = area
Watch out
r is NOT the same as the physical radius of a shape. It's the equivalent distance if area were concentrated at that radius. Small r → easy to buckle.
When to use
Column buckling (slenderness ratio λ = L/r), deflection formulas, compact sections per AISC 360
Slenderness Ratio: λ = L/r (or L/ry for weak axis)
Meaning
L = effective length of column; r = radius of gyration about buckling axis (rmin for critical)
Watch out
Use MINIMUM radius of gyration (rmin = √(Imin/A)). High λ → slender → low buckling capacity.
When to use
Column stability check per NSCP 2015, AISC 360 (Euler/Johnson buckling formulas)
Rectangular section: rx = h/√12 ≈ 0.289h (about weak axis)
Meaning
h = dimension perpendicular to weak axis
Watch out
For a rectangle, rx ≠ ry unless it's a square. Use minimum (weak axis) for buckling.
When to use
Quick check for slenderness of a column with rectangular cross-section
Common values
Value
r = h / √12 ≈ 0.289h
Symbol
rmin
Quantity
Radius of gyration: rectangle (weak axis)
Value
J = πd⁴/32
Symbol
Jcircle
Quantity
Polar moment: circle
Value
λ ≤ 200 (recommended); λ > 300 very slender
Symbol
λ
Quantity
Typical slenderness limit (NSCP/AISC)
Important facts
- For circles: Ix = Iy (by symmetry), so J = 2I.
- For rectangles: Ix ≠ Iy (one is 'strong,' one is 'weak' axis); minimum r governs column buckling.
- Slenderness ratio λ = L/r directly appears in AISC 360 buckling formulas (Euler: Fcr = π²E/(λ)²).
- NSCP 2015 §5.7.3 (Column Design) uses λ = KL/r where K is the effective length factor (K=1 for pinned ends).
- Compact/non-compact section classification per AISC 360 §B4 depends on width-to-thickness ratios, not on r directly, but r is critical for slenderness checks.
Key definitions
Polar Moment of Inertia
Example
For a circular shaft, J = πd⁴/32 governs torque capacity.
The sum of moments of inertia about two perpendicular centroidal axes; measures resistance to twisting (torsion).
Radius of Gyration
Example
A column with small r buckles easily; one with large r resists buckling.
An equivalent distance from an axis at which the entire area could be concentrated to produce the same moment of inertia. Units: mm or m.
Slenderness Ratio
Example
λ > 200 is typically very slender; λ < 50 is relatively stocky.
The dimensionless ratio λ = L/r; a measure of how slender (prone to buckling) a column is.
Diagrams to know
- Circle with Ix = Iy marked, showing J = 2I
- Rectangle with rx (weak) and ry (strong) axes, rx shown smaller
- Column with height L and cross-section showing r, slenderness ratio λ = L/r illustrated
- Buckling modes (Euler) with slenderness effect on failure load
Reactions or equations
J = Ix + Iy (perpendicular-axis theorem for centroidal axes)
Note
For circles: J = πd⁴/32. For rectangles: J = Ix + Iy; usually calculated from known Ix, Iy.
Conditions
Ix and Iy are about perpendicular centroidal axes
r = √(I/A) or equivalently I = r² A
Note
Sometimes exam asks for I given r and A; sometimes for r given I and A. Know both forms.
Conditions
Rearrangeable; useful in both directions
λ = L/r or λ_max = L/rmin (use minimum radius for buckling)
Note
Slenderness governs Euler vs Johnson buckling curve selection.
Conditions
For pinned ends (K=1); for other end conditions, use λ = KL/r per AISC/NSCP
Standard Section Centroid & Inertia Values (Quick Lookup)
Common values
Value
A ≈ 33.7 cm², Ix ≈ 1943 cm⁴, Iy ≈ 164 cm⁴, rx ≈ 8.57 cm, ry ≈ 2.21 cm
Symbol
Properties (from manual)
Quantity
IPB 200×100 (example)
Value
A ≈ 60.3 cm², Ix = Iy ≈ 2345 cm⁴ (symmetric), r ≈ 6.23 cm
Symbol
Properties (from manual)
Quantity
Hollow Box 200×200×8 (example)
Important facts
- AISC 360 (US) and Philippine structural steel manuals (e.g., Philippine National Steel Corp tables) tabulate Ix, Iy, rx, ry for standard shapes.
- Common Philippine/regional I-beam sizes: IPB (universal beam) 100×55 to 600×210 mm; values tabulated in design handbooks.
- Channels (C-shapes) and Angles (L-shapes) are asymmetric; their Ix ≠ Iy; centroid is NOT at geometric center.
- Hollow rectangular (box) and circular (pipe) sections: look up or use outer − inner formula (if concentric).
- Design codes (NSCP 2015 Section 5) reference these tabulated values directly; always confirm your section from a reputable source.
Diagrams to know
- Standard I-beam profile with centroid and principal axes labeled
- Channel (C-section) with centroid offset from geometric center
- Angle (L-section) with principal axes rotated (not aligned with edges)
- Hollow box section with outer and inner rectangles
Must remember
- PARALLEL-AXIS THEOREM: I = Ī + Ad². This is the #1 tested formula. EVERY composite section requires this. Forgetting Ad² underestimates I by 2–3×.
- CENTROID FORMULA: x̄ = ΣAᵢxᵢ / ΣAᵢ (area-weighted average). HOLES are NEGATIVE areas — subtract both A and coordinates. Get this wrong, entire solution fails.
- RECTANGLE CENTROID I: bh³/12 (CENTROID), NOT bh³/3 (base). The base formula = bh³/3. Know which axis you're using.
- TRIANGLE CENTROID: h/3 from base (NOT h/2). Memorize 1/3, not 1/2. Same for moment: Ī = bh³/36, NOT bh³/12.
- CIRCLES: Bending I = πd⁴/64. Polar moment J = πd⁴/32 (TWICE the bending I). Moment about base axis: πd⁴/64 (circles always centered at centroid anyway).
- RADIUS OF GYRATION: r = √(I/A). Governs slenderness λ = L/r (column buckling per NSCP 2015 §5.7.3). Small r → easy to buckle. Use MINIMUM radius for buckling.
- COMPOSITE SECTIONS STEP-BY-STEP: (1) Divide into simple shapes. (2) Find composite centroid. (3) Calculate Ī for each part. (4) Find d (part centroid to composite centroid). (5) Transfer: I = Ī + Ad². (6) Sum. Missing any step = wrong answer.
- AXIS OF SYMMETRY: Centroid always lies on every axis of symmetry. Use this to eliminate unknowns (e.g., if symmetric about y-axis, then x̄ is obvious).
- T-SECTION TRAP: Flange and web have different centroids. You MUST calculate the composite ȳ first, then apply parallel-axis theorem to BOTH parts (not just the web).
- HOLES ARE NEGATIVE: When calculating x̄, ȳ, A, Ī for a section with holes: subtract area, subtract coordinates, subtract (transferred) inertia. One forgotten negative term ruins the whole answer.
Last minute tips
- Always state your origin and reference axes explicitly (e.g., 'measuring from bottom-left corner'). Misaligned coordinates kill composite centroid calculations.
- For T-, I-, and L-sections: divide into TWO parts (flange/web or two legs), NOT three or four. This reduces error and matches the exam format.
- Check dimension units: if section is in mm, your Ī will be in mm⁴. If problem wants m⁴, convert by (10³)⁴ = 10¹². Don't forget unit conversion in final answer.
- The parallel-axis theorem Ad² term often exceeds Ī by 2–3× for off-center parts. If your Ad² looks small compared to Ī, re-check your calculation (likely d is too small).
- For columns (NSCP 2015 §5.7): always use rmin (= √(Imin/A)), not rmax. Using weak-axis moment of inertia ensures conservative buckling check. Using strong-axis r will underestimate buckling risk.
Comparison tables
Moment of Inertia Formulas: Centroid vs. Base (Rectangle)
| Axis | Formula | Why Different | Usage |
|---|---|---|---|
| About Centroid | Ix = bh³/12 | Shortest distance from part of section to axis is h/2 | Base for parallel-axis theorem; used to find moment about any parallel axis |
| About Base | Ix = bh³/3 | Longest distance from bottom of section is h; Ad² = (bh)(h/2)² adds 2 × centroidal moment | Rarely used directly; always convert via parallel-axis theorem in composite sections |
| Relationship | Ibase = Icentroid + A(h/2)² | bh³/3 = bh³/12 + bh(h²/4) = bh³/12 + bh³/4 = bh³/3 ✓ | This IS the parallel-axis theorem: I = Ī + Ad² |
Common Shape Centroid Locations (from reference edge)
| Shape | Centroid Location | Memo |
|---|---|---|
| Rectangle | (b/2, h/2) | At the geometric center; halfway in both directions |
| Triangle (right angle at origin) | (b/3, h/3) from right angle vertex | One-third along each leg; NOT at h/2 |
| Semicircle | 4r/(3π) ≈ 0.424r from diameter | Closer to the curved side; 4r/(3π), NOT r/2 |
| Quarter-circle | 4r/(3π) from EACH straight edge | Same formula both x and y directions from the right-angle corner |
Moment of Inertia Formulas by Shape (Centroidal Axis)
| Shape | Moment Formula | Key Note | Common Mistake |
|---|---|---|---|
| Rectangle | I = bh³/12 | b parallel to axis; h perpendicular | Using bh³/3 (base formula) instead; or confusing b and h direction |
| Triangle | I = bh³/36 | Centroid at h/3 from base; one-third of rectangle's I | Using bh³/12 (base formula); or h/2 instead of h/3 |
| Circle | I = πd⁴/64 = πr⁴/4 | Ix = Iy by symmetry; J = 2I | Using πd⁴/32 (polar, which is J not I); or confusing diameter and radius |
| Hollow Rect (concentric) | I = (b₁h₁³ − b₂h₂³)/12 | Outer − inner; both measured from shared centroid | Using if centroids do NOT coincide (must use parallel-axis theorem instead) |
Parallel-Axis Theorem Application: T-Section Example Checklist
| Step | Action | Common Error |
|---|---|---|
| 1. Divide Section | Split into simple shapes (flange, web) | Using overlapping regions instead of non-overlapping parts |
| 2. Find Composite Centroid | Use x̄ = ΣAixī / ΣAi and ȳ = ΣAiyī / ΣAi | Wrong origin; forgetting to subtract hole contributions; sign error on hole coordinates |
| 3. Calculate Part Inertias | For each part, find Ī (about part centroid) using standard formula | Using formula for wrong axis (base vs. centroid); confusing centroid location of the part |
| 4. Find Distances d | d = distance from PART centroid to COMPOSITE centroid | Measuring from wrong reference; inconsistent origin; forgetting magnitude (use |d|) |
| 5. Apply I = Ī + Ad² | For each part, calculate transferred moment; sum all parts | Forgetting the Ad² term entirely (underestimates I by 50%+); wrong sign on holes |
Diagrams
Centroids and Moments of Inertia — Core Concepts Map
Hierarchical concept map showing how centroid, moment of inertia, parallel-axis theorem, and radius of gyration interconnect. Core formulas and key triggers for each topic are displayed.
Composite Section Analysis Workflow (Parallel-Axis Theorem)
Step-by-step flowchart for solving composite section moments of inertia using the parallel-axis theorem. Highlights the critical steps, decision points, and the handling of holes. This is the exam-level procedure.