Square Pyramid

👁 DOWNLOAD Mathematica Notebook
Download Wolfram Notebook

A square pyramid is a pyramid with a square base. It is a pentahedron.

The lateral edge length 👁 e
and slant height 👁 s
of a right square pyramid of side length 👁 a
and height 👁 h
are

The corresponding surface area and volume are

The volume of a square pyramid in the special case 👁 h=a/2
can be found immediately from the cube dissection illustrated above, giving

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 👁 J_1
.

The square pyramid 👁 J_1
of edge length 👁 a
has height

and so the lateral edge length and slant height are

The surface area and volume are therefore

Consider a hemisphere placed on the base of a square pyramid (having side lengths 👁 a
and height 👁 h
). 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 👁 a/sqrt(2)
. Now find 👁 h
in terms of 👁 r
and 👁 a
. 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

Solving for 👁 h
gives

We know, however, that the hemisphere must be tangent to the sides, so 👁 r=a/2
, and

Fig. (c) shows a cross section through the center, apex, and midpoints of opposite sides. The Pythagorean theorem once again gives

We now need to find 👁 x
and 👁 y
.

But we know 👁 l
and 👁 h
, and 👁 d
is given by

so

Solving gives

so

We can now find the volume of the spherical cap as

where

so

Therefore, the volume within the pyramid is

This problem appeared in the Japanese scholastic aptitude test (Cipra 1993).

See also

Johnson Solid, Pentahedron, Pentagonal Pyramid, Pyramid, Spherical Cap, Square Pyramidal Number, Triangular Pyramid

Explore with Wolfram|Alpha

References

Cipra, B. "An Awesome Look at Japan Math SAT." Science 259, 22, 1993.

Referenced on Wolfram|Alpha

Square Pyramid

Cite this as:

Weisstein, Eric W. "Square Pyramid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquarePyramid.html

Subject classifications