

A square pyramid is a Pyramid with a Square base. If the top of the pyramid is cut off by a Plane, a
square Pyramidal Frustum is obtained. If the four Triangles of the square pyramid are
Equilateral, the square pyramid is the ``regular'' Polyhedron known as Johnson
Solid and, for side length , has height
(1) 
(2) 
(3) 
Consider a Hemisphere placed on the base of a square pyramid (having side lengths and height ). Further, let the hemisphere be tangent to the four apex edges. Then what is the volume of the Hemisphere which is interior the pyramid (Cipra 1993)?
From Fig. (a), the Circumradius of the base is . Now find in terms of and . Fig. (b) shows a
CrossSection cut by the plane through the pyramid's apex, one of the base's vertices, and the base center. This figure
gives
(4)  
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16)  
(17) 
(18) 
(19) 
See also Square Pyramidal Number
References
Cipra, B. ``An Awesome Look at Japan Math SAT.'' Science 259, 22, 1993.
© 19969 Eric W. Weisstein