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单词 Square Pyramid
释义

Square Pyramid

A square pyramid is a Pyramid with a Square base. If the top of the pyramid is cut off by a Plane, asquare Pyramidal Frustum is obtained. If the four Triangles of the square pyramid areEquilateral, the square pyramid is the ``regular'' Polyhedron known as JohnsonSolid and, for side length , has height

(1)

Using the equation for a general Pyramid, the Volumeof 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 thehemisphere be tangent to the four apex edges. Then what is the volume of the Hemisphere which is interior thepyramid (Cipra 1993)?


From Fig. (a), the Circumradius of the base is . Now find in terms of and . Fig. (b) shows aCross-Section cut by the plane through the pyramid's apex, one of the base's vertices, and the base center. This figuregives

(4)
(5)

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 Theoremonce 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
(16)
(17)

so
 
  
  
  
 (18)

Therefore, the volume within the pyramid is
 
  
 (19)

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.
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更新时间：2025/2/22 2:23:29