A pyramid is a three-dimensional polyhedron with a polygonal base and three or more triangle-shaped faces that meet above the base. The faces are the triangle sides, while the apex is the point above the base. The base is connected to the peak to form a pyramid. When the pyramid's base is in the shape of a square, the pyramid is called a square pyramid. One square base and three triangular faces make up a square pyramid. It contains 8 edges, 5 vertices, and 4 faces, in other words.
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What Is the Volume of a Square Pyramid?
The volume of a square pyramid is calculated as one-third the product of its base area and its height, expressed as volume = (1/3) × (Base Area) × (Height). This volume, quantifying the space within the pyramid, is measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), or cubic inches (in³).
A square pyramid, a type of three-dimensional geometric figure, is categorized as a pentahedron, featuring five faces. This structure includes a square base and four triangular lateral faces converging at a single point, the apex. The three main components of a square pyramid are:
- Apex: The topmost point of the pyramid.
- Base: The bottom square portion.
- Faces: The triangular sides extending from the base to the apex.
Square pyramids are evident in various objects, including the Great Pyramid of Giza and perfume bottles, illustrating their practical and historical significance.
Volume of a Square Pyramid Formula
The space contained between the five faces of a square pyramid is referred to as its volume. Knowing the base area and height of a square pyramid is all that is required to calculate its volume. The volume of a square pyramid is equal to one-third of the product of the base's area and the pyramid's height.
Formula
V = (1/3) × a2 × h
where,
a is the length of the square base,
h is the height (or altitude).
Check: Equilateral Triangle
How To Find the Volume of a Square Pyramid?
In the preceding section, we discovered that the volume of a square pyramid is calculated by multiplying the base area by the height and then by one-third. To determine the volume of a square pyramid, follow these steps:
- Step 1: Record the pyramid's dimensions, such as the base area and height, from the provided information.
- Step 2: Calculate the volume by multiplying the base area by the height and then by one-third.
- Step 3: Express the resulting volume in cubic units.
Having explored the method to calculate the volume of a square pyramid, let's clarify this concept through several solved examples.
Examples on Volume of a Square Pyramid
Problem 1. Find the volume of a square pyramid if the length of its base is 6 cm and its height is 4 cm.
Solution:
We have, a = 6 and h = 4.
Using the formula we have,
V = (1/3) × a2 × h
= (1/3) × 62 × 4
= (1/3) × 36 × 4
= 12 × 4
= 48 cm3
Problem 2. Find the volume of a square pyramid if the length of its base is 12 cm and the height is 15 cm.
Solution:
We have, a = 12 and h = 15.
Using the formula we have,
V = (1/3) × a2 × h
= (1/3) × 122 × 15
= (1/3) × 144 × 15
= 144 × 5
= 720 cm3
Problem 3. Find the length of the base of a square pyramid if its volume is 1125 cm3 and height is 15 cm.
Solution:
We have, V = 1125 and h = 15.
Using the formula we have,
V = (1/3) × a2 × h
=> 1125 = (1/3) × a2 × 15
=> 1125 = (1/3) × a2 × 15
=> 1125 = 5a2
=> a2 = 225
=> a = 15 cm
Problem 4. Find the height of a square pyramid if its volume is 1372 cm3 and base length is 14 cm.
Solution:
We have, V = 1372 and a = 14.
Using the formula we have,
V = (1/3) × a2 × h
=> 1372 = (1/3) × 14 × 14 × h
=> 1125 = (1/3) × 196 × h
=> 196 h = 4116
=> h = 21 cm
Problem 5. Find the area of the base of a square pyramid if its volume is 98 cm3 and height is 6 cm.
Solution:
We have, V = 98 and h = 6.
Using the formula we have,
V = (1/3) × a2 × h
=> 98 = (1/3) × a2 × 6
=> 98 = 2a2
=> a2 = 49 sq. cm
Practice Problems on Volume of a Square Pyramid Formula
- Calculate the volume of a square pyramid with a base side length of 8 cm and a height of 10 cm.
- A pyramid has a volume of 500 cm³ and a base length of 5 cm. Find its height.
- Determine the base length of a square pyramid with a volume of 1000 cm³ and height 20 cm.
- Find the volume of a square pyramid with base length 7 cm and height 9 cm.
- A pyramid’s volume is 375 cm³, and its height is 15 cm. What is the base length?
- Calculate the volume of a pyramid with a base side length of 3 cm and a height of 12 cm.
- If the volume of a square pyramid is 450 cm³ and the base length is 6 cm, find the height.
- Determine the height of a square pyramid with a volume of 320 cm³ and base length 8 cm.
- Find the base area of a pyramid if its volume is 600 cm³ and its height is 10 cm.
- A pyramid with base length 10 cm has a volume of 1000 cm³. What is its height?
Conclusion
The volume of a square pyramid can be efficiently calculated using the formula V=13×a2×hV=31×a2×h, where aa is the base length and hh is the height. This formula helps in quantifying the space inside the pyramid, useful in various practical and theoretical applications.