<Tr> <Td> A filled semicircle as above but with respect to an axis collinear with the base </Td> <Td> </Td> <Td> I x = π r 4 8 (\ displaystyle I_ (x) = (\ frac (\ pi r ^ (4)) (8))) I y = π r 4 8 (\ displaystyle I_ (y) = (\ frac (\ pi r ^ (4)) (8))) </Td> <Td> I x (\ displaystyle I_ (x)): This is a consequence of the parallel axis theorem and the fact that the distance between the x axes of the previous one and this one is 4 r 3 π (\ displaystyle (\ frac (4r) (3 \ pi))) </Td> </Tr> <Tr> <Td> A filled quarter circle with radius r with the axes passing through the bases </Td> <Td> </Td> <Td> I x = π r 4 16 (\ displaystyle I_ (x) = (\ frac (\ pi r ^ (4)) (16))) I y = π r 4 16 (\ displaystyle I_ (y) = (\ frac (\ pi r ^ (4)) (16))) </Td> <Td> </Td> </Tr> <Tr> <Td> A filled quarter circle with radius r with the axes passing through the centroid </Td> <Td> </Td> <Td> I x = (π 16 − 4 9 π) r 4 ≈ 0.0549 r 4 (\ displaystyle I_ (x) = \ left ((\ frac (\ pi) (16)) - (\ frac (4) (9 \ pi)) \ right) r ^ (4) \ approx 0.0549 r ^ (4)) I y = (π 16 − 4 9 π) r 4 ≈ 0.0549 r 4 (\ displaystyle I_ (y) = \ left ((\ frac (\ pi) (16)) - (\ frac (4) (9 \ pi)) \ right) r ^ (4) \ approx 0.0549 r ^ (4)) </Td> <Td> This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is 4 r 3 π (\ displaystyle (\ frac (4r) (3 \ pi))) </Td> </Tr> <Tr> <Td> A filled ellipse whose radius along the x-axis is a and whose radius along the y - axis is b </Td> <Td> </Td> <Td> I x = π 4 a b 3 (\ displaystyle I_ (x) = (\ frac (\ pi) (4)) ab ^ (3)) I y = π 4 a 3 b (\ displaystyle I_ (y) = (\ frac (\ pi) (4)) a ^ (3) b) </Td> <Td> </Td> </Tr>

Second moment of inertia formula for i beam