Circle

${\displaystyle J_{zz}=J_{xx}+J_{yy}={\frac {\pi r^{4}}{4}}+{\frac {\pi r^{4}}{4}}={\frac {\pi r^{4}}{2}}}$^[4]

where

r is the radius

This is identical to the second moment of area J_zz and is exact.

alternatively write: ${\displaystyle J={\frac {\pi D^{4}}{32}}}$^[4] where

D is the Diameter

Ellipse

${\displaystyle J\approx {\frac {\pi a^{3}b^{3}}{a^{2}+b^{2}}}}$^[5][6]

where

a is the major radius

b is the minor radius

Square

${\displaystyle J\approx \,2.25a^{4}}$^[5]

where

a is half the side length.

Rectangle

${\displaystyle J\approx \beta ab^{3}}$

where

a is the length of the long side

b is the length of the short side

${\displaystyle \beta }$ is found from the following table:

a/b	${\displaystyle \beta }$
1.0	0.141
1.5	0.196
2.0	0.229
2.5	0.249
3.0	0.263
4.0	0.281
5.0	0.291
6.0	0.299
10.0	0.312
${\displaystyle \infty }$	0.333

^[7]

Alternatively the following equation can be used with an error of not greater than 4%:

${\displaystyle J\approx {\frac {ab^{3}}{16}}\left({\frac {16}{3}}-{3.36}{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}$^[5]

where

a is the length of the long side

b is the length of the short side

Thin walled open tube of uniform thickness

${\displaystyle J={\frac {1}{3}}Ut^{3}}$^[8]

t is the wall thickness

U is the length of the median boundary (perimeter of median cross section

Circular thin walled open tube of uniform thickness

This is a tube with a slit cut longitudinally through its wall. Using the formula above:

${\displaystyle U=2\pi r}$

${\displaystyle J={\frac {2}{3}}\pi rt^{3}}$^[9]

t is the wall thickness

r is the mean radius