Right and equilateral square 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.^[3] 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.^[4] If the pyramid's apex lies on a line erected perpendicularly from the center of the square, it is called a right square pyramid, and the four triangular faces are isosceles triangles. Otherwise, the pyramid has two or more non-isosceles triangular faces and is called an oblique square pyramid.^[5]

3D model of an equilateral square pyramid

If all triangular edges are of equal length, the four triangles are equilateral, and the pyramid's faces are all regular polygons, it is an equilateral square pyramid.^[6] The dihedral angles between adjacent triangular faces are ${\displaystyle \arccos \left(-1/3\right)\approx 109.47^{\circ }}$, and that between the base and each triangular face being half of that, ${\displaystyle \arctan {\sqrt {2}}\approx 54.735^{\circ }}$.^[1] A convex polyhedron with only regular polygons as faces is called a Johnson solid, and the equilateral square pyramid is the first Johnson solid, enumerated as ${\displaystyle J_{1}}$.^[7] Like other right pyramids with a regular polygon as a base, a right square pyramid has pyramidal symmetry. For the square pyramid, this is the symmetry of cyclic group ${\displaystyle C_{4v}}$: 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. It is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base.^[1] It can be represented as the wheel graph ${\displaystyle W_{4}}$; more generally, a wheel graph ${\displaystyle W_{n}}$ is the representation of the skeleton of a ${\displaystyle n}$-sided pyramid.^[8]

Surface area and volume

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.^[9] The height ${\displaystyle h}$ of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving:^[10]

$${\displaystyle h={\sqrt {s^{2}-{\frac {l^{2}}{4}}}}={\sqrt {b^{2}-{\frac {l^{2}}{2}}}}.}$$

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:^[11]

$${\displaystyle A=4\left({\frac {1}{2}}ls\right)+l^{2}=2ls+l^{2}.}$$

In general, the volume ${\displaystyle V}$ of a pyramid is equal to one-third of the area of its base multiplied by its height.^[12] Expressed in a formula for a square pyramid, this is:^[13]

$${\displaystyle V={\frac {1}{3}}l^{2}h.}$$

Many mathematicians have 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.^[14] The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it.^[15] One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces.^[16]