How to use this calculator
Put the origin anywhere convenient, often the bottom-left corner, and describe the area as simple shapes that add up to it. Cut-outs are shapes with hole in front. The answers give the centroid from your origin, the moments of inertia about axes through the centroid, and about your x and y axes.
Formulas
Centroidx̄ = ΣAixi / ΣAi, ȳ = ΣAiyi / ΣAi (holes negative)
Parallel axisIx̄ = Σ(Ix,c + Aidy²)
Basic shapesrectangle bh³/12, circle πd⁴/64, triangle bh³/36, semicircle 0.1098r⁴
Worked example
200 × 150 mm plate with a 60 mm hole centred at (60, 100) (the default)
- A = 30 000 − 2827.4 = 27 172.6 mm².
- x̄ = (30 000 × 100 − 2827.4 × 60)/27 172.6 = 104.16 mm; ȳ = (30 000 × 75 − 2827.4 × 100)/27 172.6 = 72.40 mm.
- Ix̄ = [56.25×10⁶ + 30 000(2.60)²] − [0.636×10⁶ + 2827.4(27.60)²] = 53.66 × 10⁶ mm⁴.
Frequently asked questions
Where should I put the origin?
Anywhere. The centroidal moments of inertia don’t depend on it; only x̄, ȳ and the moments about your x and y axes do. A corner usually keeps the numbers positive.
How do I enter a W-shape or channel?
As rectangles: two flanges and a web. Fillets are ignored, so tabulated values will be a few percent higher.