## Cylindrical Wedge

The solid cut from a Cylinder by a tilted Plane passing through a Diameter of the base. It is also called a Cylindrical Hoof. Let the height of the wedge be and the radius of the Cylinder from which it is cut . Then plugging the points , , and into the 3-point equation for a Plane gives the equation for the plane as

 (1)

Combining with the equation of the Circle which describes the curved part remaining of the cylinder (and writing then gives the parametric equations of the tongue'' of the wedge as
 (2) (3) (4)

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 90° about the x-Axis using the Rotation Matrix and then by the Angle
 (5)

above the z-Axis. The transformed plane now rests in the -plane and has parametric equations
 (6) (7)

and is shown below.

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

 (8)

(and which follows immediately from the Pythagorean Theorem). The Volume of the wedge is given by
 (9)

See also Conical Wedge, Cylindrical Segment

© 1996-9 Eric W. Weisstein
1999-05-25