The volume of a pyramid is the space it occupies in a 3-dimensional plane. It is expressed in cubic units such as m3, cm3, mm3, and in3.
Formulas
The general formula to find the volume of any pyramid is:
Volume (V) = ${\dfrac{1}{3}Bh}$, here B = base area, h = height
However, there are specific formulas to calculate the volume of different pyramids due to their different shapes. They are:
Volume of a Pyramid
Let us solve some examples to understand the above concept better.
Solved Examples
Find the volume of a triangular pyramid with a base area of 64 cm2and a height of 6.3 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}Bh}$, here B = 64 cm2, h = 6.3 cm ∴ V = ${\dfrac{1}{3}\times 64\times 6.3}$ = 134.4 cm3
Finding the volume of a triangular pyramid when the BASE LENGTH, BASE HEIGHT and HEIGHT are known
Find the volume of a triangular pyramid whose height is 6 cm, base length is 7 cm, and base height is 8.5 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{6}bHh}$ , here b = 7 cm, H = 6 cm, h = 8.5 cm ∴ V = ${\dfrac{1}{6}\times 7\times 6\times 8.5}$ = 59.5 cm3
Finding the volume of an equilateral triangular pyramid when the BASE LENGTH and HEIGHT are known
Find the volume of a triangular pyramid whose height is 18 cm, and base is 4 cm each.
Solution:
As we know, The area of an equilateral triangle = ${\dfrac{\sqrt{3}}{4}b^{2}}$, here a = side of the triangle Using this formula, Volume (V) = ${\dfrac{\sqrt{3}}{12}b^{2}h}$,here b = 4 cm, h = 18 cm ∴ V = ${\dfrac{\sqrt{3}}{12}\times 4^{2}\times 18}$ = 18.47 cm3
Find the volume of a square pyramid whose height is 13 cm, and base is 5 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 5 cm, h = 13 cm ∴ V = ${\dfrac{1}{3}\times 5^{2}\times 13}$ = 108.33 cm3
Find the volume of a rectangular pyramid whose height is 8 cm, and bases are 6 cm and 9 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}lwh}$, here l = 9 cm, w = 6 cm, h = 8 cm ∴ V = ${\dfrac{1}{3}\times 9\times 6\times 8}$ = 144 cm3
Find the volume of a hexagonal pyramid whose height is 7 cm, and base is 3 cm.
Solution:
As we know, Volume (V) = ${\dfrac{\sqrt{3}}{2}b^{2}h}$ , here b = 3 cm, h = 7 cm ∴ V = ${\dfrac{\sqrt{3}}{2}\times 3^{2}\times 7}$ = 554.55 cm3
Finding the volume of a hexagonal pyramid when the BASE, APOTHEM, and HEIGHT are known
Find the volume of a hexagonal pyramid with a height of 14 cm, base of 5 cm, and apothem of 4.3 cm.
Solution:
Here we will use an alternative formula to calculate the volume. Volume (V) = abh, here a = 4.3 cm, b = 5 cm, h = 14 cm ∴ V = 4.3 × 5 × 14 = 301 cm3
Find the volume of a pentagonal pyramid with a height of 11 cm, base of 7 cm, and apothem of 4.81 cm.
Solution:
As we know, Volume (V) = ${\dfrac{5}{6}abh}$,here a = 4.81 cm, b = 7 cm, h = 11 cm ∴ V = ${\dfrac{5}{6}\times 4.81\times 7\times 11}$ = 308.64 cm3
Finding the volume of a pentagonal pyramid when the BASE and HEIGHT are known
Find the volume of a pentagonal pyramid with a height of 9.5 cm, and base of 6 cm.
Solution:
Here we will use an alternative formula to calculate the volume. Volume (V) = ${\dfrac{1.72}{3}b^{2}h}$, here b = 6 cm, h = 9.5 cm ∴ V = ${\dfrac{1.72}{3}\times 6^{2}\times 9.5}$ = 196.08 cm3
Find the volume of a given truncated pyramid.
Solution:
Volume (V) = ${\dfrac{1}{3}h^{2}\left( a^{2}+b^{2}+ab\right)}$, here a = 10 cm, b = 4 cm, h = 7.5 cm ∴ V = ${\dfrac{1}{3}\times 7.5^{2}\times \left( 10^{2}+4^{2}+10\times 4\right)}$ = 390 cm3
Find the volume of a given truncated pyramid.