Square pyramid

Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra and as the cell of a four-dimensional polytope called cubic pyramid. Square pyramidal number is a natural number that counts the number of spheres stacked into a square pyramid. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches.

Special cases

As a right pyramid

A square pyramid has five vertices, eight edges, and five faces. One face, called the base of the pyramid, is a square; the four other faces are triangles. Four of the edges make up the square by connecting its four vertices. The other four edges are known as the lateral edges of the pyramid; they meet at the fifth vertex, called the apex. If the pyramid's apex lies on a line erected perpendicularly from the center of the square, the square pyramid becomes a right pyramid, and the four triangular faces are isosceles triangles. Otherwise, the pyramid has two or more non-isosceles triangular faces.

The slant height ${\displaystyle s}$ of a right square pyramid is defined as the height of one of its isosceles triangles. It can be obtained via the Pythagorean theorem: $${\displaystyle s=\sqrt{b^2-\frac{l^2}{4}},}$$ where ${\displaystyle l}$ is the length of the triangle's base, also one of the square's edges, and ${\displaystyle b}$ is the length of the triangle's legs, which are lateral edges of the pyramid. The height ${\displaystyle h}$ of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving: $${\displaystyle h=\sqrt{s^2-\frac{l^2}{4}}=\sqrt{b^2-\frac{l^2}{2}}.}$$ A polyhedron's surface area is the sum of the areas of its faces. The surface area ${\displaystyle A}$ of a right square pyramid can be expressed as ${\displaystyle A=4T+S}$, where ${\displaystyle T}$ and ${\displaystyle S}$ are the areas of one of its triangles and its base, respectively. The area of a triangle is half of the product of its base and side, with the area of a square being the length of the side squared. This gives the expression: $${\displaystyle A=4\left(\frac{1}{2}ls\right)+l^2=2ls+l^2.}$$ In general, the volume ${\displaystyle V}$ of a pyramid is equal to one-third of the area of its base multiplied by its height. Expressed in a formula for a square pyramid, this is: $${\displaystyle V=\frac{1}{3}{l}^{2}h.}$$

Many mathematicians discovered the formula for calculating the volume of a square pyramid in ancient times. In the Moscow Mathematical Papyrus, Egyptian mathematicians demonstrated knowledge of the formula for calculating the volume of a truncated square pyramid, suggesting that they were also acquainted with the volume of a square pyramid, but it is unknown how the formula was derived. Beyond the discovery of the volume of a square pyramid, the problem of finding the slope and height of a square pyramid can be found in the Rhind Mathematical Papyrus. The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it. One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces.

Like other right pyramids with a regular polygon as a base, a right square pyramid has pyramidal symmetry, the symmetry of cyclic group ${\displaystyle C_{4\mathrm{v}}}$: the pyramid is left invariant by rotations of one-, two-, and three-quarters of a full turn around its axis of symmetry, the line connecting the apex to the center of the base; and is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base. It can be represented as the wheel graph ${\displaystyle W_4}$, meaning its skeleton can be interpreted as a square in which its four vertices connect a vertex in the center called the universal vertex. It is self-dual, meaning its dual polyhedron is the square pyramid itself.

As a Johnson solid

If all triangular edges are of equal length, the four triangles are equilateral, and the pyramid's faces are all regular polygons. The dihedral angles between adjacent triangular faces are $\arccos (-\frac{1}{3})\approx {109.47}^{\circ }$, and that between the base and each triangular face being half of that, $\arctan \left(\sqrt{2}\right)\approx {54.74}^{\circ }$. A convex polyhedron in which all of the faces are regular polygons is called a Johnson solid. Such a square pyramid is among them, enumerated as the first Johnson solid ${\displaystyle J_1}$.

Because its edges are all equal in length (that is, ${\displaystyle b=l}$), its slant, height, surface area, and volume can be derived by substituting the formulas of a right pyramid: $${\displaystyle \begin{array}{cc}s=\frac{\sqrt{3}}{2}l\approx 0.866l,& h=\frac{1}{\sqrt{2}}l\approx 0.707l,\\ A=(1+\sqrt{3})l^2\approx 2.732l^2,& V=\frac{\sqrt{2}}{6}{l}^3\approx 0.236l^3.\end{array}}$$

An equilateral square pyramid is an elementary polyhedron. This means it cannot be separated by a plane to create two small convex polyhedra with regular faces.