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) |
Using the equation for a general Pyramid, the Volume
of the ``regular'' is therefore
|
(2) |
If the apex of the pyramid does not lie atop the center of the base, then the
Slant Height is given by
|
(3) |
where is the height and is the length of a side of the base.
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
Cross-Section cut by the plane through the pyramid's apex, one of the base's vertices, and the base center. This figure
gives
so the Slant Height is
|
(6) |
Solving for gives
|
(7) |
We know, however, that the Hemisphere must be tangent to the sides, so , and
|
(8) |
Fig. (c) shows a Cross-Section through the center, apex, and midpoints of opposite sides. The Pythagorean Theorem
once again gives
|
(9) |
We now need to find and .
|
(10) |
But we know and , and is given by
|
(11) |
so
|
(12) |
Solving gives
|
(13) |
so
|
(14) |
We can now find the Area of the Spherical Cap as
|
(15) |
where
so
Therefore, the volume within the pyramid is
This problem appeared in the Japanese scholastic aptitude test (Cipra 1993).
See also Square Pyramidal Number
References
Cipra, B. ``An Awesome Look at Japan Math SAT.'' Science 259, 22, 1993.
© 1996-9 Eric W. Weisstein
1999-05-26