**Regular square pyramid**

A regular square pyramid is a right pyramid with a square base and four triangular faces.

**Mathematics**

**Keywords**

mathematics, geometry, solid geometry, solids, grouping of solids, pyramid, surface, volume, definition, motherboard, lateral surface, face, height, formula, right pyramid, tetrahedron, regular solids, oblique pyramid, vertices, faces, edges

**Related items**

### Scenes

### Formation of pyramids

Take a **polygon** and a **point** that falls **outside the plane** of the polygon. Connect the point to **all the points** on the perimeter of the polygon. The solid that is bordered by the polygon and the surface formed by the resulting line segments is called a **pyramid**. That is, a pyramid is a **conic solid** with a **polygon base**.

### Pyramids

Pyramids can be **classified** by the **base polygon**. There are pyramids with triangular, quadrilateral, pentagonal, hexagonal, etc. bases. A triangle-based pyramid is called a **tetrahedron**.

### Regular pyramids

The base of a regular pyramid is a **regular polygon** and its **lateral edges** are **equal in length**. (Therefore the lateral faces are are congruent isosceles triangles.) In regular pyramids **the base point of the height **coincides with the **centre of the base**.

### Regular square pyramid

The base of a regular square pyramid is a **regular quadrilateral**, i.e. a square. Its base edges (a) are equal in length and so are the lateral edges (b). Therefore, the lateral faces are **isosceles triangles**. In regular pyramids **the base point of the height** coincides with the centre of the square base (O).

The lateral faces of the pyramid form the **lateral surface**, which, in the case of a regular square pyramid, comprises four congruent isosceles triangles. The area of such a triangle is half the product of its base length and its height. (In other words, it is half the product of the pyramid’s base edge (a) and the slant height (l).) The **area** of the **base** is the square (a²) of the length of base edges (a). The **surface area** of the pyramid is the sum of the base area (A__{base}) and the lateral surface area (SA__{lateral})the sum of the areas of the lateral faces).

To calculate the **volume of the pyramid**, we can start with the volume of a **prism** with the same base and height as those of the pyramid. The volume of the prism is the product of its base area (A) and its height (h). The volume of the pyramid is **one third** the volume of the prism. In other words, the volume of the pyramid is one third the product of its base area and its height.

### The Egyptian pyramids

Over the course of history numerous buildings have been built with a shape of a **regular square pyramid**. The most famous ones are the **pyramids** of **Giza** in Egypt. **Pharaoh Khufu** (Cheops in Greek) had the largest of these pyramids built in the **26th century BC**. The base edges of the **Great Pyramid of Giza** were 230 m long and it measured about 147 m in height at the time of its construction.

### Related items

#### Volume of a tetrahedron

To calculate the volume of a tetrahedron we start by calculating the volume of a prism.

#### Euler's polyhedron formula

The theorem formulated by Leonhard Euler describes one of the basic properties of convex polyhedra.

#### Perimeter, area, surface area and volume

This animation presents the formulas to calculate the perimeter and area of shapes as well as the surface area and volume of solids.

#### Platonic solids

This animation demonstrates the five regular three-dimensional (or Platonic) solids, the best known of which is the cube.

#### Ratio of volumes of similar solids

This 3D scene explains the correlation between the ratio of similarity and the ratio of volume of geometric solids.

#### Solids of revolution

Rotating a geometric shape around a line within its geometric plane as an axis results in a solid of revolution.

#### Egyptian Pyramids (Giza, 26th century BC)

The Giza Necropolis is the only one of the Ancient wonders still intact.