When all of the pyramid's edges are equal in length, its triangles are all equilateral.

It is called an equilateral square pyramid, an example of a Johnson solid.

They also occur in chemistry in square pyramidal molecular structures.

Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches.

[2] Four of the edges make up the square by connecting its four vertices.

[3] 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.

of a right square pyramid is defined as the height of one of its isosceles triangles.

is the length of the triangle's legs, which are lateral edges of the pyramid.

of a right square pyramid can be similarly obtained, with a substitution of the slant height formula giving:[6]

of a pyramid is equal to one-third of the area of its base multiplied by its height.

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.

[10] The Babylonian mathematicians also considered the volume of a frustum, but gave an incorrect formula for it.

[11] One Chinese mathematician Liu Hui also discovered the volume by the method of dissecting a rectangular solid into pieces.

[12] 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.

[1] A convex polyhedron in which all of the faces are regular polygons is called a Johnson solid.

The equilateral square pyramid is among them, enumerated as the first Johnson solid

), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid:[15]

: 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.

[16] It is self-dual, meaning its dual polyhedron is the square pyramid itself.

This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces.

[19] Pyramidologists have put forward various suggestions for the design of the Great Pyramid of Giza, including a theory based on the Kepler triangle and the golden ratio.

However, modern scholars favor descriptions using integer ratios, as being more consistent with the knowledge of Egyptian mathematics and proportion.

[20] The Mesoamerican pyramids are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces.

[21] Modern buildings whose designs imitate the Egyptian pyramids include the Louvre Pyramid and the casino hotel Luxor Las Vegas.

[22] In stereochemistry, an atom cluster can have a square pyramidal geometry.

A square pyramidal molecule has a main-group element with one active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory.

For example, a tetrakis hexahedron can be constructed by attaching the base of an equilateral square pyramid onto each face of a cube.

[25] Attaching prisms or antiprisms to pyramids is known as elongation or gyroelongation, respectively.