Volume of a Cylindrical Hoof

A wedge is cut off a cylinder by a plane through a diameter of the base. The angle between the plane and the base is and the radius of the base is . Such a wedge is called a cylindrical hoof.
For a general solid , if the the cross sectional area of orthogonal to the axis has area , then the volume of the figure is given by . In the case of the hoof, it is simplest to choose the diameter as the axis. The cross section is then a triangle ABC with area , where and .
So the area is . The volume is then , where is the height.

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