Pyramid Formula

Pyramid Formula

A polyhedron with a polygonal base and triangles for sides is referred to as a pyramid. The apex, face, and base are the three essential components of any pyramid. A pyramid’s base might be of any form, such as a triangle, square, or pentagon. Faces have an isosceles triangular form. The apex of a pyramid is the point at its summit where each of the triangles intersects. The Extramarks website and mobile application both provide a detailed explanation of the Pyramid Formula for students to view.

Students can easily comprehend the concepts in a better way with the help of these solutions based on the Pyramid Formula. All necessary formulas in the solutions for the Pyramid Formula have been highlighted and mentioned in the article for students to easily find them without wasting any time.

These notes on the Pyramid Formula available on the Extramarks website and mobile application will help students clarify their doubts. The Pyramid Formula notes are extremely helpful for board examinations as they can strengthen the basics of the students. Examples citing the need for the Pyramid Formula have been discussed effectively throughout the article.

What is the Pyramid?

A pyramid is a three-dimensional shape created by connecting all of a polygon’s corners to its summit. The diagonal height measured from one of the base edges’ centres to the apex, indicated by the letter l, is known as the slant height. Height is the angle between the object’s peak and base. The symbol for height is h.

The Pyramid Formula notes are also available in Hindi for students studying under different boards. The Pyramid Formula notes have been designed and curated in a student-friendly manner, and the framework is very easy to understand for students. These solutions on the Pyramid Formula can be used for revision while also taking personal notes. The Pyramid Formula are diversified, dynamic and varied in nature, making itself a very big attraction for students.

The Hindi version of the Pyramid Formula has been made in collaboration with some of the top skilled translators of Extramarks thereby eliminating the fear of inaccuracy. Step-by-step calculations have been shown wherever necessary in the notes for the Pyramid Formula. These notes can be downloaded for offline study as well.

Types of Pyramid

The form of the bases of the various types of pyramids is used to distinguish them from one another. Here are a few examples of the many pyramidal shapes: 

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

Triangular Pyramid: A triangle pyramid is a polyhedron with three sides that meet at a location known as the apex. It has a triangular base. There are four vertices on the triangular pyramid, which can either be a regular or an irregular triangle pyramid. The only triangular pyramid with congruent equilateral triangles on each of its faces is the tetrahedron. 

Square Pyramid: A Square Pyramid is a geometric shape in three dimensions with a square base. The apex of a square pyramid is when the four triangular sides all come together.

Pentagonal Pyramid – A pyramid with a pentagonal base is referred to as a pentagonal pyramid. Five equilateral triangles and one pentagon combine to produce it. It contains 6 faces, 10 edges, and 6 vertices altogether. 

Hexagonal Pyramid: A pyramid with a hexagonal base is referred to as a hexagonal pyramid. It features six triangular lateral faces and six sides in total. A heptahedron is another name for a hexagonal pyramid. 

Right Pyramid: A particular pyramid is regarded as a right pyramid if its apex directly overlies the centre of its base. Examples of a right pyramid include square and triangular shapes.

Oblique Pyramid: An object is referred to be an oblique pyramid if its peak does not coincide with its base in the middle. The oblique pyramid’s face is incongruent.

Area of a Pyramid Formulas

• Formula for the Base Area of a Square Pyramid: b2 
• Formula for the Base Area of a Triangular Pyramid: 12 ab 
• Formula for a pentagonal pyramid’s base area: 5/2 ab 
• Formula for the Base Area of a Hexagonal Pyramid: 3AB 

The letters “a,” “b,” and “s” in the above area of the Pyramid Formula stands for the pyramid’s apothem length, base length, and slant height, respectively.

The Volume of the Pyramid

The foundation of the various forms of pyramids determines the volume of the structure. A polyhedron with only one base is called a pyramid. Therefore, the method to calculate a pyramid’s volume and surface area is based on the pyramid’s base and height. The overall capacity of the provided pyramid must be known in order to calculate the volume of the structure. One-third of the product of the base’s area and height yields the volume of a pyramid. The Pyramid Formula is,

Volume = ⅓ × Area of the base × Height

or

V = ⅓ × A × H

The pyramid’s volume may be calculated using the following units: In3, ft3, Cm3, and m3. 

Square Pyramid Volume Formula

• Formula for Square Pyramid Volume 

As is well knowledge, side2 is the formula for square area. 

Amount = a * A 

where “a” in the formula stands for a square’s side length. 

Thus, V = 13 Area of a square base is the formula for the volume of a square pyramid. Height V = 1/3 a2 H V = 1/3 b2 H 

Where V denotes the volume, a denotes the pentagonal pyramid’s apothem length, b denotes its base length, and h denotes its height.

Triangular Pyramid Volume Formula

Formula for a triangular pyramid’s volume 

As is common knowledge, a triangle’s area is calculated as follows: 

Therefore, the volume formula for a triangle pyramid is given as V = 13 Area of a triangular base. The pyramid’s height 

V = 16 abh H V = 13 (12 bh) 

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

Pentagonal Pyramid Volume Formula

As is well known, the formula for the area of a pentagon is 5/2 s a. 

Here, “s” stands for the pentagon’s side and “a” for its apothem length. 

Thus, V = 5/6 abh is the formula for the pentagonal pyramid’s volume. 

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

Hexagonal Pyramid Volume Formula

Knowing that the formula for a hexagon’s area is 3, 

3–√3/2 x² 

In this case, “x” represents a hexagonal side. 

Consequently, V = abh is the formula for the volume of a hexagonal pyramid. 

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

Surface Area of a Pyramid

• The surface area of a pyramid is calculated as the total number of square units that may adequately cover its surface. 
• The sum of the surfaces on the lateral sides of a regular pyramid is the lateral surface area. 
• The following is the general formula for a regular pyramid’s lateral surface area: 
• The formula for a pyramid’s lateral surface area is 12 pl, where “p” stands for the base’s perimeter and “l” for the slant height of the pyramid.

The Surface area of Pyramid Formulas

• Formula for a square pyramid’s surface area: 2bs + b2. 
• Formula for a triangular pyramid’s surface area: 12 ab + 3/2 bs 
• Formula for a pentagonal pyramid’s surface area: 5/2 ab plus 5/2 bs 
• Formula for a hexagonal pyramid’s surface area: 3ab + 3bs

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

1. Determine the square pyramid’s volume. It has a base length of 5 cm and a height of 9 cm. 

Solution: A = 5 cm and h = 9 cm are given. 

The formula for the volume of a square pyramid is V = 13 a2 H V = 13 (5)2 9 V = 75 cm.