Introduction
Pyramids are three-dimensional structures classified on the basis of the shape of their base. A pyramid having a triangular base is known as a triangular pyramid. Similarly, a pyramid with a square base is called a square pyramid, and so on.
The lateral faces of a pyramid meet at a common point, known as the vertex.
The height of a pyramid is the perpendicular distance from its apex to its base.
List of Formulas
Here is the formula for calculating the volume of a pyramid:
The volume of a pyramid, v = \(\frac{1}{3} \times \text{B} \times \text{h}\)
Where,
- B is the area of the base.
- h is the height of the pyramid.
Hence, the volume of a pyramid is one-third of the product of its base area and its height.
How do we use the Volume of a Pyramid formula?
To calculate the volume of a pyramid, we need to measure its base area and height. The base area can be calculated by using the area of a polygon formula.
The formulas for the volume of various types of pyramids, including the triangular pyramid, the square pyramid, and the rectangular pyramid, are shown here.
Solved Examples
Example 1: A rectangular pyramid has a height of \(\frac{27}{4}\) in. If its base area is 12 sq. in, what is its volume?
Solution:
According to the given data, height = \(\frac{27}{4}\)in
Base area = 12 sq.in
The volume of a pyramid, V = \(\frac{1}{3}\times \text{Area of the base} \times \text{height} \)
V = \(\frac{1}{3}\times 12 \times \frac{27}{4}\) [Substitute values]
V = \(\frac{1 \times 12 \times 27}{3 \times 4}\)
V = \(\frac{1 \times 12 \times 27}{12}\) [Simplify]
V = 27 [Simplify]
Therefore, the volume of the rectangular pyramid is 27 cubic inches.
Example 2: Find the height of the pyramid given below that has a volume of 36 cubic meters.
Solution:
According to the given data, volume = 36 cu.m
Base area = \(6 \times 4\) = 24 sq.m
The volume of a pyramid = \(\frac{1}{3}\times \text{Area of the base} \times \text{height}\)
\(36=\frac{1}{3}\times 24 \times x\) [Substitute values]
\(36=8 \times x\) [Simplify]
\(\frac{36}{8}=x\) [Divide both sides by 8]
\(4.5=x\) [Simplify]
Example 3: Find the side length of the base of the pyramid given below that has a volume of 605 cubic centimeters.
Solution:
According to the given data, volume = 605 cu.cm
Base area = \(x \times x = x^2\) sq.cm [Area of square formula]
The volume of a pyramid = \(\frac{1}{3}\times \text{Area of the base} \times \text{height}\)
\(\frac{1}{3}\times x^2 \times 15=605\) [Substitute values]
\(5 \times x^2=605\) [Simplify]
\(x^2=\frac{605}{5}\) [Divide both sides by 5]
\(x^2=121\)
\(x=\sqrt{121}\)
\(x=11\) cm [square root of 121 is 11]
Therefore, the side length of the base of the pyramid is 11 cm.
Example 4: The Great Pyramid of Giza is a pyramid having a square base. The base has a length of 440 cubits on each side. The pyramid stands at a height of 280 cubits. Calculate the volume of the Great Pyramid of Giza.
Solution:
According to the given data, the side length of the base = 440 cubits
Height of the pyramid = 280 cubits
Area of base = \(440 \times 440\) [Area of square formula]
= 1,93,600 sq.cubits
Volume, V = \(\frac{1}{3}\times \text{Area of the base} \times \text{height}\) [The volume of pyramid formula]
V = \(\frac{1}{3}\times \text{1,93,600} \times \text{280}\) [Substitute values]
V = \(\frac{1}{3}\times \text{54,208,000}\) [Multiply]
V= \(18,069,333.3~ \text{cubit}^3\) [Simplify]
Therefore, the volume of The Great Pyramid of Giza is 18,069,333.3 cubic cubits.
Three congruent pyramids can be formed from a cube of unit length. As a result, the volume of a pyramid is 1/3 times that of a cube. As a result, we have a 1/3 in the formula for the volume of a pyramid.
The distance from the apex along the lateral face of a pyramid is known as its slant height.
If the base of a pyramid is in the shape of a regular polygon, it is called a regular pyramid. However, if the base is in the shape of an irregular polygon, it is known as an irregular pyramid.
If the apex of a pyramid is not over the center of its base, it is known as an oblique pyramid.
A cubit is an ancient unit used to measure length. 1 cubit is generally equal to 18 inches.