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# Moment of Inertia

### Moment of Inertia of Different Shapes

Quarter-Circular Arc Centroid (x,y)= (2r/π, 2r/π) MOI= πr/2
Semi-Circular Arc Centroid (x,y)= (0, 2r/π) MOI= πr
Arc of Circle Centroid (x,y)= (rsinα/α,0) MOI= 2αr
Triangular Area Centroid (x,y)= (0,h/3) MOI= bh/3
Quarter circular area Centroid (x,y)= (4r/3π, 4r/3π) MOI= πr²/2
Quarter Elliptical Area Centroid (x,y)= (4a/3π, 4b/3π) MOI= πab/4
Semi Elliptical Area Centroid (x,y)= (0, 4b/3π) MOI= πab/2
Parabolic spandrel (y=kx²) Centroid (x,y)= (3a/4, 3h/10) MOI= ah/3
General spandrel (y=kx^n) Centroid (x,y)= ([n+1)/(n+2)]a,[(n+1)/(4n+2)]h) MOI= ah/n+1
Circular Sector Centroid (x,y)= (2rsinα/3α,0) MOI= αr²
Rectangle MOI about centroidal axis(Ix'x',Iy'y')= bh³/12, b³h/12 MOI about axes (Ixx,Iyy)= bh³/3, b³h/3 Polar moment of inertia= Ixx+Iyy
Triangle Ix'x'=bh³/36 Ixx=bh³/12
Circle Ixx= Iyy= πR^4/4 Jo= πR^4/2
Semicircle Ixx= Iyy= πR64/8 Jo= πR^4/2 Ix'x'= 0.11R64
Quarter Circle Ixx= Iyy= πR^4/16 Jo= πR^4/8
Ellipse Ixx= πab³/4 Iyy= πa³b/4 Jo= πab/4(a²+b²)

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