Pyramids

Pyramids are ancient masonry structures originally found in Egypt. The architecture is characterised by triangular sides that converge at the top. Pyramids are an important part of geometry, and in this comprehensive guide, we will learn about pyramid definition, the types of pyramid shapes, and specific pyramid examples. 

1.0What is a Pyramid?

A pyramid is described as a three-dimensional geometric shape that has a polygonal base and triangular bases that converge at a common apex. The most commonly recognised pyramid is the square pyramid, which has a square base and four triangular faces. If a pyramid has n-sided bases, then it must have n+1 faces, n+1 vertices, and 2n edges. 

2.0Types of Pyramids

Based on their shape and base, pyramids can be classified into different types in mathematics. Below are some of the major types of pyramids: 

• Square Pyramid: The base of this pyramid is the shape of a square and it has four triangular faces. 
• Triangular Pyramid: The base of the pyramid is of a triangular shape with three rectangle faces. It is the simplest type of pyramid. 
• Rectangular Pyramid: A pyramid with a rectangular base and four triangular faces. 
• Pentagonal Pyramid: These pyramids have a five-sided base with five rectangular faces. 
• Step Pyramid: These are pyramids with a series of stacked platforms resembling shapes. 
• Oblique Pyramid: It is the same as regular pyramids, where the apex is not aligned directly above the centre of the base. 

Pyramid Type	Base Shape	Number of Faces	Images
Square Pyramid	Square	5
Triangular Pyramid	Triangle	4
Rectangular Pyramid	Rectangle	5
Pentagonal Pyramid	Pentagon	6
Step Pyramid	Varies	Multiple Tiers
Oblique Pyramid	Varies	Varies

3.0Surface Area of a Pyramid

The surface area of a pyramid is defined as the sum of the areas of all its faces. In other words, the total area occupied by the surface of a pyramid is known as the surface of the pyramid. The two types of surface areas of the pyramid are lateral surface area and total surface area.

Lateral Surface Area of Regular Pyramid = (½) Pl Square units

Where P is the base perimeter of a pyramid, and I is the slant height of the pyramid.

Total Surface Area of Regular Pyramid = (½) Pl + B Square units

Where P is the base perimeter of the pyramid, I is the slant height of the pyramid, and B represents the base area of the pyramid.

Volume of a Pyramid

The volume of a pyramid refers to the space enclosed by the sides of a certain pyramid. The volume of a pyramid is therefore: 

V = ⅓ A × H

Where, A = area and H = height

4.0Mathematical Concepts Related to Pyramids

Here are some mathematical concepts related to the pyramid example and types of pyramids. 

• The Golden Ratio and Pyramids: The dimensions of the Great Pyramid of Gaza are closely associated with the golden ratio, which is the aesthetically pleasing proportion in architecture and design.
• Fractal Pyramidal Structures: Fractal pyramidal structures like the Sierpiński Pyramid show that the pyramid shape can be repeated infinitely, demonstrating a self-similarity in mathematical models.
• Pyramids in Vector Geometry: Pyramids can be studied using vector calculus, especially in physics and engineering applications. 
• Pyramids in Coordinate Geometry: Using 3D coordinated geometry, the equation of pyramidal structures can be derived for computer graphics, simulation, and virtual modelling. 

5.0Solved Examples

Example: A square pyramid has a base side length of 6 cm and a slant height of 10 cm. Find its total surface area.

TSA = Base Area + Lateral Surface Area

TSA = s² + 2s x I

Here, s = the side of the base, and I = the slant height.

TSA = 6² + 2 x 6 x 10

TSA = 36 + 120 = 156 cm²

Example: Find the volume of a square pyramid with a base side length of 8 cm and a height of 12 cm.

The volume (V) of a pyramid is given by:

⅓ x A x H 

Where A = area and H = height.

The base area would be: s² = 8² = 64²

V = ⅓ x 64 x 12 = 768/3 = 256 cm³. 

Example: A triangular pyramid has a triangular base with a base length of 5 cm and a height of 4 cm. Each triangular face has a height of 7 cm. Find its total surface area.

Total Surface Area (TSA) = Base Area + Lateral Surface Area

The Base Area is: ½ x base x height = ½ x 5 x 4 = 10 cm²

Since it is a triangular pyramid, it has 3 triangular faces.

Each lateral face has the following: 

Area of one triangular face = ½ x base x slant height = ½ x 5 x 7 = 17.5 cm²

Since there are three faces, 

Lateral Surface Area = 3 x 17.5 = 52.5 cm²

Total Surface Area = 10 + 52.5 = 62.5 cm².

Example: Find the volume of a triangular pyramid with a base area of 30 cm² and a height of 10 cm.

The volume of a pyramid is ⅓ x base area x height 

V = ⅓ x 30 x 10 = 100 cm³.

6.0Conclusion

Pyramids are more than just ancient relics. They are fundamental in geometry, physics, and architectural principles. Even after ages, pyramid shapes continue to influence science and design. Understanding the intricacies of a pyramid only helps us understand our heritage and how to preserve it.