# Pyramid

A pyramid is a polyhedron, whose base can be any polygon and whose lateral faces are triangles with a common vertex (**apex** of the pyramid).

The **height** of the pyramid is the measure of the perpendicular line segment from the apex to the base.

The **apothem** of a regular pyramid is the height of one of its lateral faces.

#### Unfold of a Pyramid

## Types of Pyramids

### Regular Pyramid

The base of a regular pyramid is a regular polygon and its faces are equally sized.

### Irregular Pyramid

The base of an irregular pyramid is an irregular polygon, and as a result, its faces are not equally sized.

### Convex Pyramid

### Concave Pyramid

### Right Pyramid

A right pyramid has isosceles triangles as its lateral faces and its apex lies directly above the midpoint of the base.

### Oblique Pyramid

An oblique pyramid does not have all isosceles triangles as its sides.

### Types of the Pyramids by their Base

### Triangular Pyramid

### Square Pyramid

The base is a square.

### Pentagonal Pyramid

### Hexagonal Pyramid

#### Calculation of the Apothem of a Pyramid

The apothem of a pyramid can be calculated if the height of the pyramid and apothem of the base are known. Apply the Pythagorean theorem and calcuate the hypotenuse for the shaded right triangle:

#### Calculation of the Lateral Edge of a Pyramid

The lateral edge of the pyramid can be calucated if the height of the pyramid and the radius of the base (or radius of the circumscribed circle) are known. Apply the Pythagorean theorem and calcuate the hypotenuse of the shaded right triangle.

### Lateral Area of a Pyramid

### Surface Area of a Pyramid

### Volume of a Pyramid

#### Examples

Calculate the lateral area, surface area and volume of a square pyramid whose base edge is 10 cm and its height is 12 cm.

Calculate the lateral area, surface area and volume of a hexagonal pyramid whose base edge is 16 cm and its lateral edge is 28 cm.