Home / United States / Math Classes / Formulas / Volumes of 3D Figures – Pyramids Formulas
Volume is the amount of space that an object occupies. A pyramid is a threedimensional figure with a polygonshaped base and triangular lateral faces meeting at a common point. Here we will focus on the formula used to calculate the volume of pyramids....Read MoreRead Less
Pyramids are threedimensional 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.
Type  Diagram  Properties 
Square pyramid 
 
Triangular pyramid 

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,
Hence, the volume of a pyramid is onethird of the product of its base area and its height.
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.
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.