What is the Square Pyramid Volume Formula?
A square pyramid is a three-dimensional geometrical shape with four triangular lateral faces and a square base. The lateral faces are connected to the square base and meet at a common point called the apex of the pyramid.
So a square pyramid has three main parts:
- The apex that is the top point or vertex of the pyramid
- The square shaped bottom of the pyramid that is known as the base
- The triangle-shaped lateral faces that are called faces
The image shows a square pyramid with five faces in total.
The measure of the space enclosed by the five faces of a square pyramid is known as its volume. The volume of a square pyramid can be determined by using the formula:
Volume of a Square Pyramid, \( V=(\frac{1}{3}\times b^2\times h) \)
where,
- V refers to the volume of the square pyramid
- b refers to the side length of the square base and
- h refers to the height of the pyramid
[Note: The volume of the pyramid like any other solid is measured in cubic units or unit\( ^3 \).]
Rapid Recall
Solved Examples
Example 1: A food stall in a movie theater sells French fries in a packaging box that has a shape similar to a square pyramid with a base of side length 4 inches and a height of 10 inches. Find the volume of the packaging box.
Solution:
As mentioned, the shape of the packaging box is a square pyramid. So we can use the formula for volume of a square pyramid to calculate its volume.
That is,
Volume of Square Pyramid, \( V=\frac{1}{3}\times b^2\times h \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{1}{3}\times (4)^2\times 10 \) [Substitute 4 for b and 10 for h]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{1}{3}\times 16\times 10 \) [Square of 4]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{160}{3} \) [Multiply]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=53.33 \) [Divide]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\approx 53 \)
Therefore, the volume of the square pyramid shaped packaging box is approximately 53 cubic inches.
Example 2: If the volume of a square pyramid that has a height of 8 inches is 40cubic inches, then what would be the length of an edge of its base side?
Solution:
As provided, the volume of the square pyramid is,
V = 40 cubic inches
Its height is 8 inches
The side length of its square base can be determined by applying the formula for the volume of a square pyramid, that is,
Volume of Square Pyramid, \( V=\frac{1}{3}\times b^2\times h \)
\( 40=\frac{1}{3}\times b^2\times 8 \) [Substitute 40 for V and 8 for h]
\( 40\times 3=b^2\times 8 \) [Multiply both sides by 3]
\( 120=b^2\times 8 \) [Simplify]
\( \frac{120}{8}=b^2 \) [Divide both sides by 8]
\( 15=b^2 \) [Simplify]
\( b=3.87 \) [Take positive square root on both sides]
Hence, the base length of the given square pyramid is 3.87 inches.
Example 3:
Chloe traveled to Egypt to see the pyramids there. Her tour guide claimed that the pyramids can hold 1,960,000 cubic units of water and have a square base of side length of 120 feet while describing the history of the pyramids. Chloe wondered what the height of the pyramid could be. Help Chloe find the answer.
Solution:
Given, the volume of the pyramid is 1960000 cubic feet and the square base side length is 120 feet.
So, to help Chloe find the height of the pyramid, we apply the formula for the volume of a square pyramid.
That is,
Volume of Square Pyramid, \( V=\frac{1}{3}\times b^2\times h \)
\( 1960000=\frac{1}{3}\times (120)^2\times h \) [Substitute 1960000 for V and 120 for b]
\( 1960000=\frac{1}{3}\times 14400\times h \) [Simplify]
\( 1960000=\frac{14400}{3}\times h \)
\( 1960000=4800\times h \) [Simplify further]
\( \frac{1960000}{4800}=h \) [Divide both sides by 4800]
\( h=408.33 \) [Simplify]
Thus, the height of the pyramid is 408.33 feet.
Cubic units are used to measure the volume of a square pyramid. For example cubic meters, cubic centimeters, liters (l) and so on.
The three main parts of a square pyramid are the apex, the square base, and the faces.
Pyramids of Egypt, toys, tents, mobile towers, watermelon slices, wet floor signs are some real-life examples of square pyramids.
The height of a square pyramid is the perpendicular distance from the apex to the center of the square base.
A square pyramid has 8 edges.
A square pyramid has 5 vertices.