Pyramid

The structure of a pyramid is 3-D or 3-dimensional, in nature. The shape of the pyramid is formed when corners of a polygon are joined to the apex of the central point. The diagonal height comes from the distance between the apex to the center of any one of the base edges. It is a slant height. A perpendicular distance from the base of the pyramid to the apex point is termed the height of the pyramid. The sides of the pyramid essentially are formed of triangles. However, the base may or may not be formed of a triangle. This introduces the various types of pyramids. These varieties include- a square pyramid, a triangular pyramid, and a pentagonal pyramid. However, the base of a regular pyramid is considered to be square. 

Pyramid Formula

Any pyramid includes the apex, faces/sides, and the base. It is the base of the pyramid which determines whether it is a triangular, square, or pentagonal pyramid. When the base is a square, it is a square pyramid. When the base is triangular, it is a triangular pyramid. Similarly, when the base is pentagonal, it is a pentagonal pyramid. Moreover, a pyramid triangle is an isosceles (two sides of the triangle are equal in its length). This establishes that the slant height is equal all around the pyramid. 

These are the core concepts that will aid in understanding the formulas. These formulas include- a pyramid’s surface area and a regular pyramid’s total surface area. 

The surface area or lateral surface area of a pyramid is derived from the following formula:-

• L.S.A. = P×L× ½  

Where, P= Perimeter of the pyramid’s base 

               L= Slant height

A regular pyramid’s total surface area is derived from the following formula:-

• T.S.A. = ½ × P × L + B

Where, P= Perimeter of the pyramid’s base

               L= Slant height

               B= Area of the pyramid’s base

• Square pyramid’s surface area= 2×L×B×B2

Where B= base length

        L= slant length

        B2= Area of a square

• Triangular pyramid’s surface area= ½ × B× A + 3/2 × L × B

Where B = Base length

               L=Slant Height

               A= Apothem length (the distance between the center of the base and the middle point of the base’s sides) 

• Pentagonal pyramid’s surface area= 5/2 (B×A + L ×B)

Where B= Base length

               A= Apothem length

               L= Slant Height

Pyramid Volume

The pyramid volume is the sum of the surface areas of the pyramid base and the lateral faces of that particular pyramid. 

Therefore, the pyramid volume can be derived from the following formula:-

• ½ × height × Base area
• Square pyramid’s volume = 1/3 × H × B2

               Where H= height of the pyramid (perpendicular from the base to the apex)

                             B= Base length

• Triangular pyramid’s volume= 1/6 × B × A × H

Where B= Base length

               A= Apothem length

               H= Height

• Pentagonal pyramid’s volume = 5/6 × B × A × H

Where B= Base length

               A= Apothem length

               H= height

Pyramid of Numbers

A pyramid of numbers is a particular type of quantitative aptitude or quantitative reasoning question. It involves a pyramid, which is more of a triangle filled with a series of numbers. The numbers are arranged in a certain manner to make distinct patterns or number series. The question asks to find out those patterns or explain those series. A variety of questions can be framed from the pyramid of numbers. This makes the topic much more diverse. To solve questions of the pyramid of numbers one must understand the number system well. 

Conclusion

The topic of pyramids tried is vast, in terms of the questions it can present. However, formulas are a very important tool to answer such questions. It has been explained what pyramids are and what are its various parts. Surface area, total surface area, and volume of pyramids have also been elaborated. The pyramid of numbers is also an important part of quantitative aptitude. The main understanding of the pyramid of numbers has been established. FAQs are presented to clear queries that commonly come up. The section also contains problem questions that shall help understand the concepts and application of formulas better.