Pyramid Formula

A pyramid is defined as a polyhedron that has a base, and three or more triangular faces that meet at a point known as the apex. These triangular sides of the pyramid are often called lateral faces to differentiate them from the base of the pyramid. The base of the pyramid can be any polygon, but mostly a pyramid has a square base. All the faces of the pyramid are usually triangular in shape. If the base of the pyramid is a regular polygon, then the triangles of the pyramid will be congruent (similar shape and size), and isosceles (two sides with similar length). In this article, we will discuss what a pyramid is, different types of a pyramid, the area of the pyramid, volume of a pyramid, the surface area of a pyramid etc.

Types of Pyramid

The different types of the pyramid are differentiated from each other on the basis of the shape of their bases. Some of the different types of the pyramid are as follows:

• Triangular Pyramid

• Square Pyramid

• Pentagonal Pyramid

• Right Pyramid

• Oblique Pyramid

Let us now discuss the different types of pyramids briefly. 

Triangular Pyramid - A triangular pyramid is a  polyhedron with a triangular base and the three faces of the triangular pyramid meet at a point known as the apex. The triangular pyramid has 4 vertices and it can be either a regular triangular pyramid or an irregular triangular pyramid. The tetrahedron is the only triangular pyramid having congruent equilateral triangles for each of its faces. 

Square Pyramid - A square pyramid is a three-dimensional geometrical figure having a square base. All the four triangular sides of a square pyramid meet at one single point known as the apex.  

Pentagonal Pyramid - A pentagonal pyramid is a pyramid having a pentagonal base. It is formed by 5 equilateral triangles and 1 pentagon. It has a total of 6 faces, 10 edges, and 6 vertices. 

Hexagonal Pyramid - A hexagonal pyramid is a pyramid having a hexagonal base. It has a total of 6 sides and 6 triangular lateral faces. A hexagonal pyramid is also known as a heptahedron.

Right Pyramid - If the apex of the pyramid directly lies over the center of the base then the particular pyramid is considered as a right pyramid. The square pyramid, triangular pyramid are examples of a right pyramid. 

Oblique Pyramid - If the apex of the pyramid does not lie over the center of the base then the particular pyramid is considered as an Oblique pyramid. The face of the oblique pyramid is not congruent. 

Area of a Pyramid Formulas

• Base Area of a square pyramid formula =  b²

• Base Area of a triangular pyramid formula = ½ ab 

• Base Area of a pentagonal pyramid formula =  5/2 ab 

• Base Area of a hexagonal pyramid formula = 3ab 

In the above area of pyramid formulas, 'a' represents the apothem length of the pyramid, 'b' represents the base length of the pyramid and 's' represents the slant height of the pyramid.

The Volume of the Pyramid

The volume of the pyramid relies upon the base of the different types of a pyramid. A pyramid is a polyhedron having only one base. Hence, the formula to determine the volume and the surface area of a pyramid are based on the base and height of the pyramid. To determine the volume of a pyramid,it is necessary to know the total capacity of the given pyramid. The pyramid volume formula is derived by one-third of the product of the area of the base to its height. Hence, the pyramid volume formula is

Volume = ⅓ × Area of the base × Height

or

V = ⅓ × A × H

The volume of the pyramid can be measured in the following units:

In³, ft³, Cm³, m³

Square Pyramid Volume Formula

As we know the formula for the area of a square is side²

Area  = a * a

Where ‘a’ in the formula represents the length of the sides of a square.

Hence, the square Pyramid Volume Formula is given as :

V = ⅓ × Area of a square base ×Height

V = ⅓ × a² ×H

V = ⅓  b²  H

Where V is the volume, a is the apothem length of the pentagonal pyramid, b is the base length of the pentagonal pyramid and h is the height of the pentagonal pyramid

Triangular Pyramid Volume Formula

As we know the area of a triangle is ½ * base * height

Hence, triangle pyramid volume formula is given as :

V = ⅓ × Area of a triangular base ×Height of the pyramid

V = ⅓ × (½ bh) ×H

V = ⅙ abh

Where V is the volume, a is the apothem length of the pentagonal pyramid, b is the base length of the pentagonal pyramid and h is the height of the pentagonal pyramid.

Pentagonal Pyramid Volume Formula

As we know the area of a pentagon formula is 5/2 × s ×a

Here, ‘s’ is the side of the pentagon and ‘a’ is the apothem length of the pentagon.

Hence, the pentagonal pyramid volume formula is given as :

V = 5/6 abh

Where V is the volume, a is the apothem length of the pentagonal pyramid, b is the base length of the pentagonal pyramid and h is the height of the pentagonal pyramid.

Hexagonal Pyramid Volume Formula

As, we know the area of a hexagon formula is 3

3–√3/2 x²

Here, ‘x’ is the side of the hexagon.

Hence, the hexagonal pyramid volume formula is given as 

V = abh

Where V is the volume, a is the apothem length of the pentagonal pyramid, b is the base length of the pentagonal pyramid and h is the height of the pentagonal pyramid.

Surface Area of a Pyramid

• The total number of square units that appropriately covers the surface of a pyramid is determined as the surface area of a pyramid.

• The lateral surface area of a regular pyramid is the total of the areas of its lateral faces.

• The general formula for the lateral surface area of a regular pyramid is given as 

• The lateral surface area of a pyramid formula - ½ pl, 

• where ‘p’ is the perimeter of the base and ‘l’ is the slant height of the pyramid.

• The total surface area of a regular pyramid is the total of the areas of its lateral faces and its base.

• The general formula for the total surface area of a regular pyramid is given as:

• The Total surface area of a pyramid formula =  ½ pl + B, 

• Where ‘p’ is the perimeter of the base and ‘l’ is the slant height of the pyramid and B is the area of the base.

The Surface area of Pyramid Formulas

• The surface area of a square pyramid formula - 2bs + b²

• The surface area of a triangular  pyramid formula - ½ ab + 3/2 bs

• The surface area of a pentagonal pyramid formula - 5/2 ab + 5/2 bs

• The surface area of a hexagonal pyramid formula - 3ab + 3bs

In the above surface area of pyramid formulas, a represents the apothem length of the pyramid, b represents the base length of the pyramid and s represents the slant height of the pyramid.

Solved Example

1. Calculate the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 8 inches and the slant height is 5 inches.

Solution: As we know, the formula for the lateral surface area of a regular pyramid is ½ pl, where ‘p’ is the perimeter of the base and ‘l’ is the slant height of the pyramid.

The perimeter of the base is the total of all the sides of  the pyramid’

P = 3(8) = 24 inches

L.S.A. = ½(24)(5) 

= 60 inches²

2. Calculate the volume of the square pyramid of base length 5 cm and height 9 cm.

Solution: Given , a = 5 cm and h = 9 cm

The square pyramid volume formula = V = ⅓  a²  H

V = ⅓ × (5)² × 9

V = 75 cm

Quiz questions 

1. Non-right pyramids are known as

1. Oblique pyramid

2. Pentagon based pyramid

3. Hexagon based pyramid

4. Tetrahedron based pyramid

2. A geometrical figure with a polygon base and vertex is known as :

1. Diamond

2. Pyramid

3. Triangle

4.  Dice