In geometry, a square pyramid is a pyramid with a square base and four triangular lateral faces. We can find different parameters for a square pyramid, such as surface area and volume. We know that the volume of a pyramid is dependent on the base area of that pyramid. The below figure shows the shape of a square pyramid. It has a square base, four triangular or lateral faces connected at a vertex opposite to the base.
In this article, you will learn the formula for the volume of a square pyramid, derivation of the formula, and solved examples on the volume of the square pyramid.
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Volume of a Square Pyramid Formula
Consider a square pyramid with altitude or height “h”, and length of the edge of the base “a”.
As we know, the volume of any pyramid = (⅓) × Base area × Height
So, the formula for the volume of a square pyramid = (⅓) × Area of square × Height
= (⅓) × a2 × h
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Volume of a Square Pyramid = (⅓)a2h |
Volume of a Square Pyramid with Slant Height
It is possible to find the volume of a square pyramid using slant height, which means we can derive the formula for the volume of a square pyramid without height. This can be done as follows.
The above figure shows a square pyramid with the edge of the base “a”, height “h”, and slant height “l”.
By Pythagoras theorem,
l2 = h2 + (a/2)2
h2 = l2 – (a/2)2
Substituting this equation in the volume formula, we get;
Volume of a square pyramid =
Also, try: Volume of a Square Pyramid Calculator
Volume of Frustum of a Square Pyramid
When a plane cuts the square pyramid parallel to its base, we can get the frustum of a square pyramid. When the top of the square pyramid is removed, the lower part of the square pyramid is called the frustum of a square pyramid. This can be shown as:
Now, consider the height of the frustum of the square pyramid as “H” and the lower base as “Base 1” with edge s1 and the upper base as “Base 2” with edge s2.
Area of the lower base = S1 = (s1)2
Area of the upper base = S2 = (s2)2
Thus, the volume of the frustum of a square pyramid = (H/3) [S1 + S2 + √(S1S2)]
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Solved Examples
Example 1:
Find the volume of a square pyramid whose height is 14 units and the edge of the base is 9 units.
Solution:
Given,
Height = h = 14 units
Edge of the base of square pyramid = a = 9 units
Volume = (⅓)a2h
= (⅓) × 9 × 9 × 14
= 378 cubic units
Therefore, the volume of the given square pyramid is 378 cubic units.
Example 2:
If the volume of a square pyramid is 294 cm3 and the height of the pyramid is 18 cm, find the measure of the edge of the base.
Solution:
Let “a” be the edge of the base of a square pyramid.
Given,
Height = h = 18 cm
Volume of a square pyramid = 294 cm3
⇒ (⅓) a2h = 294
⇒ (⅓) × a2 × 18 = 294
⇒ a2 = (294 × 3)/18
⇒ a2 = 49
⇒ a = ±7
We know that the measure of an edge cannot be negative.
So, the measure of the edge of the base = 7 cm.
Practice Questions
- Find the volume of a square pyramid whose slant height is 5 cm and the edge of the base is 6 cm.
- If the volume of a square pyramid is 1620 cubic units and the measure of the edge of the base is 18 units, find the height of the pyramid.
- Calculate the volume of the frustum of a square pyramid with height 16 cm, and base areas 81 cm2 and 36 cm2, respectively.
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