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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 
), so the three-point form of the plane gives the equation
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(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=intz(x)y(x)dx
=2int_(R-b)^R(h(x-R+b))/bsqrt(R^2-x^2)dx
=h/(6b)[2sqrt((2R-b)b)(3R^2-2ab+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/Inline30.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/Inline43.svg) |
(16)
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 |  | ![(2hR)/b[a+(b-R)phi]](/images/equations/CylindricalWedge/Inline46.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 wegde, 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/Inline66.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).
While the centroid of the general cylindrical wedge is complicated for 
,
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(29)
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for the semicircular cylindrical wedge, the centroid is given by
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(30)
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giving
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(31)
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(32)
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(33)
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