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A square pyramid is a pyramid with a square base. It is a pentahedron.
The lateral edge length 
and slant height 
of a right square pyramid of side length 
and height 
are
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(1)
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(2)
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The corresponding surface area and volume are
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(3)
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(4)
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The volume of a square pyramid in the special case 
can be found immediately from the cube dissection illustrated
above, giving
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(5)
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If the four triangles of the square pyramid are equilateral, so that all edges of the square pyramid have the same lengths, then the right square
pyramid is the polyhedron known as Johnson solid 
.
The square pyramid 
of edge length 
has height
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(6)
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and so the lateral edge length and slant height are
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(7)
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(8)
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The surface area and volume are therefore
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(9)
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(10)
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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 that
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
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(11)
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(12)
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so the slant height is
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(13)
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Solving for 
gives
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(14)
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We know, however, that the hemisphere must be tangent to the sides, so 
,
and
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(15)
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Fig. (c) shows a cross section through the center, apex, and midpoints of opposite sides. The Pythagorean theorem once again gives
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(16)
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We now need to find 
and 
.
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(17)
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But we know 
and 
,
and 
is given by
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(18)
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so
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(19)
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Solving gives
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(20)
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so
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(21)
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We can now find the volume of the spherical cap as
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(22)
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where
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(23)
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(24)
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so
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(25)
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Therefore, the volume within the pyramid is
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(26)
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(27)
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This problem appeared in the Japanese scholastic aptitude test (Cipra 1993).
