The torsion constant is a geometrical property of a beam's cross-section which determines the relationship between angle of twist and applied torque.

For a beam of uniform cross-section along its length:

$theta\; =\; frac\{TL\}\{JG\}$

$theta$ is the angle of twist in radians
T is the applied torque
L is the beam length
J is the torsion constant
G is the modulus of rigidity of the material

For non-circular cross-sections, there are no exact analytical equations for finding J. Approximate solutions have been found for many shapes.

Examples for specific cross-sectional shapes

Circle

$J\; =\; frac\{pi\; r^4\}\{2\}$
r is the radius
This is identical to the polar moment of inertia and is exact.

Hollow concentric circular tube

$J\; =\; frac\{pi\}\{2\}\; left\; (r\_o^4\; -\; r\_i^4\; right\; )$
$r\_o$ is the outer radius
$r\_i$ is the inner radius
This is identical to the polar moment of inertia and is exact.

Square

$J\; approx\; frac\{a^4\}\{7.10\}$
a is the side length

Rectangle

$J\; =beta\; a\; b^3$
a is the length of the long side
b is the length of the short side
$beta$ is found from the following table:

a/b	$beta$
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
$infty$	0.333

Alternatively the following equation can be used with an error of not greater than 4%:
$J\; approx\; a\; b^3\; left\; (frac\{1\}\{3\}-0.210\; frac\{b\}\{a\}\; left\; (1-\; frac\{b^4\}\{12a^4\}\; right\; )\; right\; )$

Thin walled closed tube of uniform thickness

$J\; =\; frac\{4A^2t\}\{U\}$
A is the mean of the areas enclosed by the inner and outer boundaries
t is the wall thickness
U is the length of the median boundary

Thin walled open tube of uniform thickness

$J\; =\; frac\{1\}\{2\}U\; t^3$
t is the wall thickness
U is the length of the median boundary

Circular thin walled open tube of uniform thickness

This is a tube with a slit cut longitudinally through its wall.
$J\; =\; frac\{2\}\{3\}\; pi\; r\; t^3$
t is the wall thickness
r is the mean radius
This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.

References