The angular twist 
of a shaft with given cross section is given by
 |
(1)
|
(Roark 1954, p. 174), where 
is the twisting moment (commonly measured in units of inch-pounds-force),
is the length (inches), 
is the modulus of rigidity (pounds-force per square inch),
and 
(sometimes also denoted 
) is the torsional rigidity multiplier for a given geometric
cross section (inches to the fourth power). Note that the quantity 
is sometimes denoted 
(e.g., Timoshenko and Goodier 1951, p. 264).
Values of 
are known exactly only for a small number of cross sections, and in closed form for
even fewer. The following table lists approximate values for some common shapes (Timoshenko
and Goodier 1951, pp. 258-280; Roark 1954, pp. 174-179).
| cross section |  approx | OEIS |
| circle | 1.570796... | A019669 |
| equilateral triangle | 0.021650... | A180317 |
| half-disk | 0.297556... | A180310 |
| isosceles right triangle | 0.026089... | A180314 |
| quarter-disk | 0.0825... | |
| sliced disk | 0.878055... | A180311 |
| square | 0.140577... | A180309 |
Closed forms are known for the annulus
 |
(2)
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(Roark 1954, p. 175), circle
 |
(3)
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(Roark 1954, p. 174), ellipse
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(4)
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(Timoshenko and Goodier 1951, p. 263-265; Roark 1954, p. 174), equilateral triangle
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(5)
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(Timoshenko and Goodier 1951, p. 265-267; Roark 1954, p. 175), and half-disk and slit full disk (i.e., circular sector from 0 to 
)
 |  |  |
(6)
|
 |  |  |
(7)
|
(E. Weisstein, Aug. 27, 2010; given approximately by Saint-Venant 1878; Timoshenko and Goodier 1951, p. 263-265; Roark 1954, p. 174).
Exact solutions expressed as sums (with no known closed form) are known for the rectangle and square
 |  | ![(a^3b)/3[1-(192)/(pi^5)a/bsum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1)b)/(2a))]](/images/equations/TorsionalRigidity/Inline20.svg) |
(8)
|
 |  | ![(a^4)/3[1-(192)/(pi^5)sum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1))/2)]](/images/equations/TorsionalRigidity/Inline23.svg) |
(9)
|
(Timoshenko and Goodier 1951, pp. 275-277), isosceles right triangle
![K_(isos. rt. tri.)=a^4[1/(12)-(16)/(pi^5)sum_(n=1)^infty1/((2n-1)^5)coth((pi(2n-1))/2)]](/images/equations/TorsionalRigidity/NumberedEquation6.svg) |
(10)
|
(Galerkin 1919; correcting the typo 1/2 for 1/12), and circular sector
 |
(11)
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where
![f(r,psi)=-r^2[1-(cos(2psi))/(cosalpha)]+(16a^2alpha^2)/(pi^3)sum_(n=1,3,5,...)(-1)^((n+1)/2)(r/a)^(npi/alpha)(cos((npipsi)/alpha))/(n(n+(2alpha)/pi)(n-(2alpha)/pi))](/images/equations/TorsionalRigidity/NumberedEquation8.svg) |
(12)
|
(Saint-Venant 1878; Greenhill 1879; Dinnik, and Föppl and Föppl 1928; Timoshenko and Goodier 1951, pp. 278-280).
