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A cylindrical wedge, also called a cylindrical hoof or cylindrical ungula, is a wedge cut from a cylinder by
slicing with a plane that intersects
the base of the cylinder. The volume
of a cylindrical wedge can be found by noting that the plane
cutting the cylinder passes through the three points
illustrated above, with 
and 
. The illustration has 
. The three-point form of the plane
gives the equation
 |  | ![2a[h(R-b-x)+bz]](/images/equations/CylindricalWedge/Inline6.svg) |
(1)
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(2)
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Solving for 
gives
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(3)
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Here, the value of 
is given by
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(4)
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(5)
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The volume is therefore given as an integral over rectangular areas along the x-axis,
![V=int_(R-b)^R2z(x)sqrt(R^2-x^2)dx
=2int_(R-b)^R(h(x-R+b))/bsqrt(R^2-x^2)dx
=h/(6b)[2sqrt((2R-b)b)(3R^2-2Rb+b^2)-3piR^2(R-b)+6R^2(R-b)sin^(-1)((R-b)/R)].](/images/equations/CylindricalWedge/NumberedEquation2.svg) |
(6)
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Using the identities
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(7)
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(8)
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(9)
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(10)
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gives the equivalent alternate forms
 |  | ![h/(3b)[a(3R^2-a^2)+3R^2(b-R)phi]](/images/equations/CylindricalWedge/Inline32.svg) |
(11)
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(12)
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(Harris and Stocker 1998, p. 104). This simplifies in the case of 
to
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(13)
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The lateral surface area can be found from
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(14)
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where 
is simply 
with 
,
so
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(15)
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 |  | ![(2hR)/b{sqrt(2bR-b^2)+(b-R)×[1/2pi+tan^(-1)((b-R)/(sqrt(2bR-b^2)))]}](/images/equations/CylindricalWedge/Inline45.svg) |
(16)
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 |  | ![(2hR)/b[a+(b-R)phi]](/images/equations/CylindricalWedge/Inline48.svg) |
(17)
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(18)
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(Harris and Stocker 1998, p. 104).
A special case of the cylindrical wedge which might be called a semicircular cylindrical wedge, is a wedge passing through a diameter
of the base (so that 
).
Let the height of this wedge be 
and the radius of the cylinder
from which it is cut be 
. Then plugging the points 
, 
, and 
into the 3-point equation for a plane
gives the equation for the plane
as
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(19)
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Combining with the equation of the circle that describes the curved part remaining of the cylinder
(and writing 
)
then gives the parametric equations of the
"tongue" of the wedge as
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(20)
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(21)
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(22)
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for ![t in [0,R]](/images/equations/CylindricalWedge/Inline68.svg)
.
To examine the form of the tongue, it needs to be rotated into a convenient plane.
This can be accomplished by first rotating the plane of
the curve by 
about the x-axis
using the rotation matrix 
and then by the angle
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(23)
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above the z-axis. The transformed plane now rests in the 
-plane
and has parametric equations
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(24)
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(25)
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and is shown below.
The length of the tongue (measured down its middle) is obtained by plugging 
into the above equation
for 
,
which becomes
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(26)
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(and which follows immediately from the Pythagorean theorem).
As determined from the case of the general cylindrical wedge, the volume of the semicircular cylindrical hoof is given by
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(27)
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and the lateral surface area by
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(28)
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The volume was found by Gregory of St. Vincent (1647).
The centroid of the general cylindrical wedge is given by
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(29)
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which evaluates to
 |  | ![h/(12bV)[3R^4phi-a(R-b)(3R^2+2a^2)]](/images/equations/CylindricalWedge/Inline82.svg) |
(30)
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(31)
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(32)
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(G. M. Tynan, pers. comm., Oct. 15, 2008). For the semicircular cylindrical wedge, the centroid is given by
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(33)
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giving
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(34)
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(35)
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(36)
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