Pyramid Formula - Definition, Types, Derivation, Examples

A Pyramid is a geometric polyhedron characterized by having a base, typically in the shape of a polygon and three or more triangular faces that converge at a single point called the apex. The triangular faces of the pyramid are often referred to as lateral faces to distinguish them from the base. While the base of a pyramid can take the form of any polygon, pyramids are frequently constructed with a square base. In most cases, all the faces of the pyramid are triangular. When the base of the pyramid is a regular polygon, the triangular faces of the pyramid are not only triangular but also congruent (having the same shape and size) and isosceles (with two sides of equal length).

In this article, we will explore the concept of pyramids, various types of pyramids, methods for calculating the area and volume of pyramids, and details related to pyramid surface area, among other topics.

Types of Pyramid

Pyramids come in various forms, each distinguished by the shape of their bases. Here is a brief overview of some different types of pyramids:

• Triangular Pyramid
• Square Pyramid
• Pentagonal Pyramid
• Right Pyramid
• Oblique Pyramid

Triangular Pyramid (Tetrahedron): A triangular pyramid has a triangular base and three triangular lateral faces that meet at a single apex. Each triangular face is an equilateral triangle, meaning all sides and angles are of equal length.

Square Pyramid (Pyramid with Square Base): A square pyramid has a square base and four triangular lateral faces that meet at an apex. The triangular faces are typically isosceles triangles, meaning they have two sides of equal length.

Pentagonal Pyramid: A pentagonal pyramid has a pentagonal (five-sided) base and five triangular lateral faces that converge at an apex. The triangular faces are isosceles triangles.

Right Pyramid: A right pyramid is one in which the apex is directly above the center of the base. This means that the height of the pyramid is perpendicular to the base.

Oblique Pyramid: An oblique pyramid is one where the apex is not directly above the center of the base. In this case, the height is not perpendicular to the base.

These are some of the main types of pyramids, but pyramids can be created with bases of various polygonal shapes, resulting in diverse forms and structures. Understanding the type of pyramid is important when calculating its volume, surface area, and other geometric properties.

Area of a Pyramid Formulas

• Base Area of a square pyramid formula =  b ^²
• 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 a pyramid depends on the shape and area of its base. To calculate the volume of a pyramid, you need to know the area of its base and its height. The formula for determining the volume of a pyramid is derived by taking one-third of the product of the area of the base and the height. Therefore, the pyramid volume formula is expressed as:

Volume = (1/3) × Area of the base × Height

This formula can also be represented as:

V = (1/3) × A × H

The volume of a pyramid can be measured in various units, including cubic inches (In³), cubic feet (ft³), cubic centimetres (cm³), and cubic meters (m³). The specific unit of measurement depends on the units used for the area of the base and the height. For consistency, it's important to ensure that the units are compatible before calculating the volume of the pyramid.

Square Pyramid Volume Formula

For a square-based pyramid, the formula for volume is indeed as follows:

V = (1/3) × (Area of the square base) × Height

V = (1/3) × (a ^² ) × H

In this formula:

V represents the volume of the square-based pyramid.

'a' is the length of the side of the square base.

'H' is the height of the pyramid, which is the perpendicular distance from the base to the apex.

Pentagonal Pyramid Volume Formula

For a pentagonal pyramid, which has a pentagonal (five-sided) base, the formula for volume can be expressed as:

V = (5/12) × (b ^² ) × H

In this formula:

V represents the volume of the pentagonal pyramid.

'b' is the length of one of the sides of the pentagonal base.

'H' is the height of the pyramid, which is the perpendicular distance from the base to the apex.

Please note the difference in the formula for pentagonal pyramids compared to square-based pyramids. The base shape and the number of sides affect the volume formula used for each type of pyramid.

Triangular Pyramid Volume Formula

For a triangular pyramid, the formula for volume is indeed as follows:

V = (1/3) × (Area of the triangular base) × Height

V = (1/3) × (1/2) × b × h × H

V = (1/6) × b × h × H

In this formula:

V represents the volume of the triangular pyramid.

'b' is the length of the base of the triangular base.

'h' is the height of the triangular base.

'H' is the height of the pyramid, which is the perpendicular distance from the base to the apex.

Hexagonal Pyramid Volume Formula

The formula for calculating the volume of a hexagonal pyramid is as follows:

V = (2/3) × (Area of the hexagonal base) × Height

Where:

V is the volume of the hexagonal pyramid.

Area of the hexagonal base refers to the area of the hexagonal (six-sided) base.

Height is the perpendicular distance from the base to the apex of the pyramid.

Keep in mind that to use this formula, you need to know the measurements of the hexagonal base (side length of apothem) and the height of the pyramid. You can calculate the area of the hexagonal base using the appropriate formula for a regular hexagon, and then use that area in the volume formula. The formula assumes that the base is a regular hexagon with all sides and angles equal.

Surface Area of a Pyramid

The surface area of a pyramid is defined as the total number of square units required to cover its surface adequately. In the case of a regular pyramid, which has a polygonal base and triangular lateral faces converging at a single apex, its surface area can be divided into two components:

Lateral Surface Area: The lateral surface area of a regular pyramid encompasses the combined areas of its lateral faces. The general formula for calculating the lateral surface area of such a pyramid is given by:

Lateral Surface Area = (1/2) × Perimeter of the Base (p) × Slant Height (l)

In this formula:

p represents the perimeter of the base.

l is the slant height of the pyramid, which is the height of a lateral face.

Total Surface Area: The total surface area of a regular pyramid encompasses not only the lateral faces but also the area of its base. The general formula for calculating the total surface area is expressed as:

Total Surface Area = (1/2) × Perimeter of the Base (p) × Slant Height (l) + Area of the Base (B)

In this formula:

p stands for the perimeter of the base.

l represents the slant height of the pyramid.

B denotes the area of the base, which depends on the shape of the base (e.g., square, triangular, etc.).

These formulas are instrumental in determining the surface area of a regular pyramid, which is essential for various applications, including material estimation and understanding the pyramid's external characteristics.

The Surface area of Pyramid Formulas

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

Pyramid Formula Examples

Example 1: Calculate the Volume of a Square Pyramid

Suppose you have a square pyramid with a base side length (a) of 5 cm and a height (H) of 8 cm. To find the volume (V), you can use the formula for a square pyramid:

V = (1/3) × (a ^² ) × H

V = (1/3) × (5 cm) ^² × 8 cm

V = (1/3) × 25 cm ^² × 8 cm

V = (1/3) × 200 cm ^³

V = 66.67 cm ^³

The volume of the square pyramid is approximately 66.67 cubic centimetres.

Example 2: Calculate the Lateral Surface Area of a Triangular Pyramid

Let's say you have a triangular pyramid with a base perimeter (p) of 12 cm, slant height (l) of 7 cm, and no given base area. To calculate the lateral surface area (LSA), you can use the formula for the lateral surface area of a triangular pyramid:

LSA = (1/2) × p × l

LSA = (1/2) × 12 cm × 7 cm

LSA = 42 cm ^²

The lateral surface area of the triangular pyramid is 42 square centimetres.

Example 3: Calculate the Total Surface Area of a Hexagonal Pyramid

Suppose you have a hexagonal pyramid with a base perimeter (p) of 18 cm, slant height (l) of 9 cm, and base area (B) of 108 cm ^² . To calculate the total surface area (TSA), you can use the formula for the total surface area of a regular pyramid:

TSA = (1/2) × p × l + B

TSA = (1/2) × 18 cm × 9 cm + 108 cm ^²

TSA = 81 cm ^² + 108 cm ^²

TSA = 189 cm ^²

The total surface area of the hexagonal pyramid is 189 square centimetres.

These examples demonstrate how to apply pyramid formulas to calculate the volume and surface area of different types of pyramids, such as square, triangular, and hexagonal pyramids.