Pyramid

Components of a 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

Unfold of a Pyramid


Types of Pyramids

Regular Pyramid

Regular Pyramid


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

Irregular Pyramid

Irregular Pyramid


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

Convex Pyramid

Convex Pyramid


A convex pyramid has a convex polygon as its base.

Concave Pyramid

Concave Pyramid


A concave pyramid has a concave polygon as its base.

Right 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

Oblique Pyramid


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

Types of the Pyramids by their Base

Triangular Pyramid

Triangular Pyramid


The base is a triangle.

Square Pyramid

Quadrangular Pyramid


The base is a square.

Pentagonal Pyramid

Pentagonal Pyramid


The base is a pentagon.

Hexagonal Pyramid

Hexagonal Pyramid


The base is a hexagon.


Calculation of the Apothem of a Pyramid

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:

Apothem of a Pyramid Formula

Apothem of a Pyramid Formula

Calculation of the Lateral Edge of a Pyramid

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 Edge of a Pyramid Formula

Lateral Edge of a Pyramid Formula

Lateral Area of a Pyramid

Lateral Area of a Pyramid Formula

Lateral Area of a Pyramid Formula

Surface Area of a Pyramid

Surface Area of a Pyramid Formula

Volume of a Pyramid

Volume of a Pyramid Formula


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.

Pyramid Problem

Pyramid Exercise

Pyramid Operations

Pyramid Operations

Pyramid Operations

Pyramid Operations

Pyramid Solution

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.

Hexagonal Pyramid

Pyramid Exercise

Pyramid Operations

Pyramid Solution

Hexagonal Pyramid

Pyramid Exercise

Pyramid Operations

Pyramid Solution

Hexagonal Pyramid

Pyramid Exercise

Pyramid Operations

Pyramid Solution