Section Properties: Centroid and Moment of Inertia Formulas

Area, centroid and second moment of area (moment of inertia) formulas for 27 cross sections, from basic shapes to structural steel profiles. The moment of inertia about the centroidal axes is what you need for bending stresses and deflections.

Open any section for its full set of properties (inertia about the base, plastic modulus, torsional constant), or press Calculate to get the values for your dimensions.

Basic shapes

Section	Area	Centroid	Inertia about x̄	Inertia about ȳ
Square	$$A = l^2$$	$$\bar{x} = \frac{l}{2}$$ $$\bar{y} = \frac{l}{2}$$	$$\bar{I}_x = \frac{l^4}{12}$$	$$\bar{I}_y = \frac{l^4}{12}$$	Calculate →
Diamond Square	$$A = l^2$$	$$\bar{x} = \frac{l}{\sqrt{2}}$$ $$\bar{y} = \frac{l}{\sqrt{2}}$$	$$\bar{I}_x = \frac{l^4}{12}$$	$$\bar{I}_y = \frac{l^4}{12}$$	Calculate →
Rectangle	$$A = b h$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{h}{2}$$	$$\bar{I}_x = \frac{b h^3}{12}$$	$$\bar{I}_y = \frac{b^3 h}{12}$$	Calculate →
Rotated Rectangle	$$A = b h$$	$$\bar{x} = \frac{b \cos(\theta) + h \sin(\theta)}{2}$$ $$\bar{y} = \frac{b \sin(\theta) + h \cos(\theta)}{2}$$	$$\bar{I}_x = \frac{b h}{12} (b^2 \sin^2(\theta) + h^2 \cos^2(\theta))$$	$$\bar{I}_y = \frac{b h}{12} (b^2 \cos^2(\theta) + h^2 \sin^2(\theta))$$	Calculate →
Triangle	$$A = \frac{b h}{2}$$	$$\bar{x} = \frac{a + b}{3}$$ $$\bar{y} = \frac{h}{3}$$	$$\bar{I}_x = \frac{b h^3}{36}$$	$$\bar{I}_y = \frac{bh(a^2-ab+b^2)}{36}$$	Calculate →
Isosceles Trapezoid	$$A = \frac{h(a + b)}{2}$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{h(2a + b)}{3(a + b)}$$	$$\bar{I}_x = \frac{h^3(a^2 + 4ab + b^2)}{36(a + b)}$$	$$\bar{I}_y = \frac{h(a + b)(a^2 + b^2)}{48}$$	Calculate →
General Trapezoid	$$A = \frac{h(a + b)}{2}$$	$$\bar{x} = \frac{a^2 + ab + 2ac + b^2 + bc}{3(a + b)}$$ $$\bar{y} = \frac{h(2a + b)}{3(a + b)}$$	$$\bar{I}_x = \frac{h^3(a^2 + 4ab + b^2)}{36(a + b)}$$	$$\bar{I}_y = I_y - \bar{x}^2 A$$	Calculate →

Circular shapes

Section	Area	Centroid	Inertia about x̄	Inertia about ȳ
Circle	$$A = \pi r^2$$	$$\bar{x} = r$$ $$\bar{y} = r$$	$$\bar{I}_x = \frac{\pi r^4}{4}$$	$$\bar{I}_y = \frac{\pi r^4}{4}$$	Calculate →
Half Circle	$$A = \frac{\pi r^2}{2}$$	$$\bar{x} = 0$$ $$\bar{y} = \frac{4r}{3\pi}$$	$$\bar{I}_x = \left(\frac{\pi}{8} - \frac{8}{9\pi}\right) r^4$$	$$\bar{I}_y = \frac{\pi r^4}{8}$$	Calculate →
Quarter Circle	$$A = \frac{\pi r^2}{4}$$	$$\bar{x} = \frac{4r}{3\pi}$$ $$\bar{y} = \frac{4r}{3\pi}$$	$$\bar{I}_x = \left(\frac{\pi}{16} - \frac{4}{9\pi}\right) r^4$$	$$\bar{I}_y = \left(\frac{\pi}{16} - \frac{4}{9\pi}\right) r^4$$	Calculate →
Circle Sector	$$A = \frac{\theta r^2}{2}$$	$$\bar{x} = \frac{4r \sin(\theta/2)}{3\theta}$$ $$\bar{y} = 0$$	$$\bar{I}_x = \frac{r^4(\theta - \sin\theta)}{8}$$	$$\bar{I}_y = I_y - \bar{x}^2 A$$	Calculate →

Hollow and thin-walled sections

Section	Area	Centroid	Inertia about x̄	Inertia about ȳ
Hollow Rectangle	$$A = b h - b_i h_i$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{h}{2}$$	$$\bar{I}_x = \frac{b h^3}{12} - \frac{b_i h_i^3}{12}$$	$$\bar{I}_y = \frac{b^3 h}{12} - \frac{b_i^3 h_i}{12}$$	Calculate →
Thin-Walled Rectangle	$$A = b h - b_i h_i$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{h}{2}$$	$$\bar{I}_x = \frac{b h^3}{12} - \frac{b_i h_i^3}{12}$$	$$\bar{I}_y = \frac{b^3 h}{12} - \frac{b_i^3 h_i}{12}$$	Calculate →
Hollow Circle	$$A = \pi (R^2 - r^2)$$	$$\bar{x} = R$$ $$\bar{y} = R$$	$$\bar{I}_x = \frac{\pi}{4} (R^4 - r^4)$$	$$\bar{I}_y = \frac{\pi}{4} (R^4 - r^4)$$	Calculate →
Thin-Walled Circle	$$A = \pi (R^2 - r^2)$$	$$\bar{x} = R$$ $$\bar{y} = R$$	$$\bar{I}_x = \frac{\pi}{4} (R^4 - r^4)$$	$$\bar{I}_y = \frac{\pi}{4} (R^4 - r^4)$$	Calculate →

Regular polygons

Section	Area	Centroid	Inertia about x̄	Inertia about ȳ
Hexagon	$$A = \frac{3\sqrt{3}}{2}l^2$$	$$\bar{x} = l$$ $$\bar{y} = \frac{\sqrt{3}}{2}l$$	$$\bar{I}_x = \frac{5\sqrt{3}}{16}l^4$$	$$\bar{I}_y = \frac{5\sqrt{3}}{16}l^4$$	Calculate →
Octagon	$$A = 2(\sqrt{2}+1) l^2$$	$$\bar{x} = \frac{\sqrt{2}+1}{2} l$$ $$\bar{y} = \frac{\sqrt{2}+1}{2} l$$	$$\bar{I}_x = \frac{11+8\sqrt{2}}{12} l^4$$	$$\bar{I}_y = \frac{11+8\sqrt{2}}{12} l^4$$	Calculate →

Structural profiles

Section	Area	Centroid	Inertia about x̄	Inertia about ȳ
I-Beam	$$A = td +2s(b -t)$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{d}{2}$$	$$\bar{I}_x = \frac{bs^3}{6} +\frac{bs(d-s)^2}{2} +\frac{t(d-2s)^3}{12}$$	$$\bar{I}_y = \frac{b^3s}{6} +\frac{t^3(d -2s)}{12}$$	Calculate →
Tapered I-Beam	$$A = dt + (b-t)(g+s)$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{d}{2}$$	$$\bar{I}_x = \frac{d^3 t}{12} + \frac{(b - t)(3d^2(g + s) - 4d(g^2 + gs + s^2) + 2(g + s)(g^2 + s^2))}{12}$$	$$\bar{I}_y = \frac{d t^3}{12} + \frac{(b-t)((b + t)^2(g + s) + 2(gt^2 + sb^2))}{24}$$	Calculate →
Unequal I-Beam	$$A = dt+s_1(b_1-t)+s_2(b_2-t)$$	$$\bar{x} = \frac{\text{max}(b_1,b_2)}{2}$$ —	—	$$\bar{I}_y = \frac{dt^3 + s_1(b_1^3-t^3) + s_2(b_2^3 - t^3)}{12}$$	Calculate →
T-Beam	$$A = bs+t(d-s)$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{bs(2d-s)+t(d-s)^2}{2A}$$	$$\bar{I}_x = I_x - \bar{y}^2 A$$	$$\bar{I}_y = \frac{b^3s}{12} + \frac{t^3(d-s)}{12}$$	Calculate →
Tapered T-Beam	$$A = bs + \frac{(d-s)(t_1+t_2)}{2}$$	$$\bar{x} = \frac{b}{2}$$ $$\bar{y} = \frac{3bs(2d-s) + (d-s)^2(t_1+2t_2)}{6A}$$	$$\bar{I}_x = I_x - \bar{y}^2 A$$	$$\bar{I}_y = \frac{b^3s}{12} + \frac{(d-s)(t_1+t_2)(t_1^2+t_2^2)}{48}$$	Calculate →
Square Channel	$$A = 2bs + t(d-2s)$$	$$\bar{x} = \frac{2b^2s + t^2(d-2s)}{4bs + 2t(d-2s)}$$ $$\bar{y} = \frac{d}{2}$$	$$\bar{I}_x = I_x - \bar{y}^2A$$	$$\bar{I}_y = I_y - \bar{x}^2A$$	Calculate →
Tapered Channel	$$A = bd - (b-t)(d-n-s)$$	$$\bar{x} = \frac{2b^2(n+2s) + 2b(n-s)t + (3d-4n-2s)t^2}{6A}$$ $$\bar{y} = \frac{d}{2}$$	$$\bar{I}_x = I_x - \bar{y}^2A$$	—	Calculate →
Square Angle	$$A = a^2 - (a-t)^2$$	$$\bar{x} = \frac{a^2 + at - t^2}{2(2a - t)}$$ $$\bar{y} = \frac{a^2 + at - t^2}{2(2a - t)}$$	$$\bar{I}_x = I_x - \bar{y}^2 A$$	$$\bar{I}_y = I_y - \bar{x}^2 A$$	Calculate →
Rectangular Angle	$$A = t(a + b - t)$$	$$\bar{x} = \frac{at+b^2-t^2}{2(a+b-t)}$$ $$\bar{y} = \frac{a^2+bt-t^2}{2(a+b-t)}$$	$$\bar{I}_x = I_x - \bar{y}^2 A$$	$$\bar{I}_y = I_y - \bar{x}^2 A$$	Calculate →
Z-Beam	$$A = t(2b + d - 2t)$$	$$\bar{x} = b-\frac{t}{2}$$ $$\bar{y} = \frac{d}{2}$$	$$\bar{I}_x = \frac{t(2b(3d^2-6dt+4t^2) + (d-2t)^3)}{12}$$	$$\bar{I}_y = \frac{t((2b-t)^3+t^2(d-t))}{12}$$	Calculate →

Notation and axes

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