A wedge is 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

(1)

Solving for gives

(2)

The volume is therefore given as an integral over rectangular areas along the x-axis,

(3)

Using the identities

			(4)
			(5)
			(6)
			(7)

gives the equivalent alternate forms

			(8)
			(9)

(Harris and Stocker 1998, p. 104). This simplifies in the case of to

(10)

The lateral surface area can be found from

(11)

where is simply with , so

			(12)
			(13)
			(14)
			(15)

(Harris and Stocker 1998, p. 104).

A special case of the cylindrical wedge, also called a cylindrical hoof, is a wedge passing through a diameter of the base (so that ). Let the height of the 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

(16)

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

			(17)
			(18)
			(19)

for . 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

(20)

above the z-axis. The transformed plane now rests in the -plane and has parametric equations

			(21)
			(22)

and is shown below.

The length of the tongue (measured down its middle) is obtained by plugging into the above equation for , which becomes

(23)

(and which follows immediately from the Pythagorean theorem).

As already determined from the case of the general cylindrical wedge, the volume of the cylindrical hoof is given by

(24)

and the lateral surface area by

(25)

While the centroid of the general cylindrical wedge is complicated for ,

(26)

for the cylindrical hoof with , the centroid is given by

(27)

giving

			(28)
			(29)
			(30)

Harris, J. W. and Stocker, H. "Obliquely Cut Circular Cylinder" and "Segment of a Cylinder." §4.6.3-4.6.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 103-104, 1998.

Kern, W. F. and Bland, J. R. "Truncated Prism (or Cylinder)." §31 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 81-83 and 127, 1948.

CITE THIS AS:

Weisstein, Eric W. "Cylindrical Wedge." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CylindricalWedge.html