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MI of Some Standard Bodies
Moment of Inertia of Some Standard Bodies About Different Axes
Body 
Axis of Rotation 
Figure 
Moment of Inertia 
k 
k^{2}/R^{2} 
Ring 
About an axis passing through C.G. and perpendicular to its plane 
MR^{2} 
R 
1 

Ring 
About its diameter 
½ MR^{2} 
R / (2)^{1/2} 
½ 

Ring 
About a tangential axis in its own plane 
3/2 MR^{2} 
(3/2)^{1/2} R 
3/2 

Ring 
About a tangential axis perpendicular to its own plane 
2MR^{2} 
(2)^{1/2} R 
2 

Disc 
About an axis passing through C.G. and perpendicular to its plane 
½ MR^{2} 
R / (2)^{1/2} 
½ 

Disc 
About its Diameter 
¼ MR^{2} 
R/2 
¼


Disc 
About a tangential axis in its own plane 
5/4 MR^{2} 
((5)^{1/2} / 2) R 
5/4 

Disc 
About a tangential axis perpendicular to its own plane 
3/2 MR^{2} 
(3/2)^{1/2} R 
3/2 

Annular disc inner radius = R_{1} and outer radius = R_{2} 
Passing through the centre and perpendicular to the plane 
(M/2) [R_{1}^{2} + R_{2}^{2}] 
 
 

Annular disc 
Diameter 
(M/4) [R_{1}^{2} + R_{2}^{2}] 
 
 

Annular disc 
Tangential and Parallel to the diameter 
(M/4) [5 R_{1}^{2} + R_{2}^{2}] 
 
 

Annular disc 
Tangential and perpendicular to the plane 
(M/2) [3 R_{1}^{2} + R_{2}^{2}] 
 
 

Solid cylinder 
About its own axis 
½ MR^{2} 
R / (2)^{1/2} 
½ 

Solid cylinder 
Tangential(Generator) 
3/2 MR^{2} 
(3/2)^{1/2} R^{2} 
3/2 

Solid cylinder 
About an axis passing through its C.G. and perpendicular to its own axis 
M[(L^{2}/12) + (R^{2}/4)] 
((L^{2}/12) + (R^{2}/4))^{1/2} 


Solid cylinder 
About the diameter of one of faces of the cylinder 
M[(L^{2}/3) + (R^{2}/4)] 
((L^{2}/3) + (R^{2}/4))^{1/2} 


Cylindrical shell 
About its own axis 
MR^{2} 
R 
1 

Cylindrical shell 
Tangential(Generator) 
2MR^{2} 
(2)^{1/2} R 
2 

Cylindrical shell 
About an axis passing through its C.G. and perpendicular to its own axis 
M[(L^{2}/12) + (R^{2}/2)] 
((L^{2}/12) + (R^{2}/2))^{1/2} 


Cylindrical shell 
About the diameter of one of faces of the cylinder 
M[(L^{2}/3) + (R^{2}/2)] 
((L^{2}/3) + (R^{2}/2))^{1/2} 


Hollow cylinder with inner radius = R_{1} and outer radius = R_{2} 
Axis of cylinder 
(M/2) (R_{1}^{2} + R_{2}^{2}) 



Solid Sphere 
About its diametric axis 
2/5 MR^{2} 
(2/5)^{1/2} R 
2/5 

Solid sphere 
About a tangential axis 
7/5 MR^{2} 
(7/5)^{1/2} R 
7/5 

Spherical shell 
About its diametric axis 
2/3 MR^{2} 
(2/3)^{1/2} R 
2/3 

Spherical shell 
About a tangentialAxis 
5/3 MR^{2} 
(5/3)^{1/2} R 
5/3 

Hollow sphere of inner radius R_{1} and outer radius R_{2}

About its diametric axis 
2/5 M [(R_{2}^{5} – R_{1}^{5}) / (R_{2}^{3} – R_{1}^{3})] 



Hollow sphere 
Tangential 




Long thin rod 
About on axis passing through its centre of mass and perpendicular to the rod. 
ML^{2}/12 
L / (12)^{1/2} 


Long thin rod 
About an axis passing through its edge and perpendicular to the rod 
ML^{2}/3 
L / (3)^{1/2} 


Rectangular lamina of length l and breadth b 
Passing through the centre of mass and perpendicular to the plane 
(M/12)[l^{2} + b^{2}] 



Rectangular lamina 
Tangential perpendicular to the plane and at the midpoint of breadth 
(M/12)[4l^{2} + b^{2}] 



Rectangular lamina 
Tangential perpendicular to the plane and at the midpoint of length 
(M/12)[l^{2} + 4b^{2}] 



Rectangular parallelopiped length l, breadth b, thickness t 
Passing through centre of mass and parallel to(i) Length (x)(ii) breadth (z)(iii) thickness (y) 
(i) M[b^{2} + t^{2}] / 12 (ii) M[l^{2} + t^{2}] / 12 (iii) M[b^{2} + l^{2}] / 12 



Rectangular parallelepiped length l, breath b, thickness t 
Tangential and parallel to(i) length (x)(ii) breadth (y)(iii) thickness(z) 
(i) (M/12)[3l^{2} + b^{2} + t^{2}] (ii) (M/12) [l^{2} + 3b^{2} + t^{2}] (iii) (M/12) [l^{2} + b^{2} + 3t^{2}] 



Elliptical disc of semimajor axis = a and semiminor axis = b 
Passing through CM and perpendicular to the plane 
M/4 [a^{2} + b^{2}] 



Solid cone of radius R and height h 
Axis joining the vertex and centre of the base 
3/10 MR^{2} 



Equilateral triangular lamina with side a 
Passing through CM and perpendicular to the plane 
Ma^{2}/6 



Right angled triangular lamina of sides a, b, c 
Along the edges 
(i) Mb^{2}/6 (ii) Ma^{2}/6 (iii) M/6 [ a^{2}b^{2} / a^{2 }+ b^{2}] 

