The volume of a pyramid is the space enclosed inside a three-dimensional pyramid shape. A pyramid has a polygon base and triangular faces that meet at a common point called the apex. The height of the pyramid is the perpendicular distance from the apex to the baseand the volume is measured in cubic units such as cm³m³or in³.
Relation Between Slant Height and Height
In a pyramidthe triangle formed by slant height (s)vertical height (h)and half of the base side (x/2) is a right-angled triangle.
Using the Pythagorean theorem:
s² = h² + (x/2)²
Derivation
Consider a pyramid and a prism having the same base area and height. If the pyramid is filled with water and poured into the prismit fills only one-third of the prism.
This shows that the volume of a pyramid is one-third the volume of a prism with the same base and height.
So,
Volume of Prism = Base Area × Height
Therefore,
Volume of Pyramid (V) = (1/3) × Base Area × Height
Formulas for Different Types of Pyramids
The volume of a pyramid is given by: Volume = (1/3) × Base Area × Height
The base is a polygonso its area is calculated using the respective polygon formulas and then substituted into the above formula.
Below are the volume formulas of different types of pyramids.
Sample Problems
Problem 1: What is the volume of a square pyramid if the sides of the base are 6 cm each and the height of the pyramid is 10 cm?
Solution:
Given
- Length of Side of Base of Square Pyramid = 6 cm
- Height of Pyramid = 10 cm
Volume of Square Pyramid (V) = 1/3 × Area of square base × Height
Area of square base = a2 = 62 = 36 cm2
V = 1/3 × (36) ×10 = 120 cm3
Hencevolume of the given square pyramid is 120 cm3.
Problem 2: What is the volume of a triangular pyramid whose base area and height are 120 cm2 and 13 cmrespectively?
Solution:
Given
- Area of Triangular Base = 120 cm2
- Height of Pyramid = 13 cm
Volume of a Triangular Pyramid (V) = 1/3 × Area of Triangular Base × Height
V = 1/3 × 120 × 13 = 520 cm3
Hencevolume of the given triangular pyramid = is 520 cm3
Problem 3: What is the volume of a triangular pyramid if the length of the base and altitude of the triangular base are 3 cm and 4.5 cmrespectivelyand the height of the pyramid is 8 cm?
Solution:
Given
- Height of Pyramid = 8 cm
- Length of Base of Triangular Base = 3 cm
- Length of Altitude of Triangular Base = 4.5 cm
Area of Triangular Base (A) = 1/2 b × h = 1/2 × 3 × 4.5 = 6.75 cm2
Volume of Triangular Pyramid (V) = 1/3 × A × H
V = 1/3 × 6.75 × 8 = 18 cm3
Hencevolume of the given triangular pyramid is 18 cm3
Problem 4: What is the volume of a rectangular pyramid if the length and width of the rectangular base are 8 cm and 5 cmrespectivelyand the height of the pyramid is 14 cm?
Solution:
Given
- Height of Pyramid = 14 cm
- Length of Rectangular Base (l) = 8 cm
- Width of Rectangular Base (w) = 5 cm
Area of Rectangular Base (A) = l × w = 8 × 5 = 40 cm2
We have,
Volume of Rectangular Pyramid (V) = 1/3 × A × H
V = 1/3 × 40 × 14 = 560/3 = 186.67 cm3
Hencevolume of the given rectangular pyramid is 186.67 cm3.
Problem 5: What is the volume of a hexagonal pyramid if the sides of the base are 8 cm each and the height of the pyramid is 15 cm?
Solution:
Given,
Side of hexagon (a) = 8 cm
Height (h) = 15 cmArea of hexagonal base:
A = (3√3 / 2) × a²
A = (3√3 / 2) × (8)²
A = (3√3 / 2) × 64
A = 96√3 cm²Volume of pyramid:
V = (1/3) × A × h
V = (1/3) × 96√3 × 15
V = 480√3 cm³
Problem 6: What is the volume of a pentagonal pyramid if the base area is 150 cm2 and the height of the pyramid is 11 cm?
Solution:
- Area of Pentagonal Base = 150 cm2
- Height of Pyramid = 11 cm
Volume of Pentagonal Pyramid (V) = 1/3 × Area of Pentagonal Base × Height
V = 1/3 × 150 × 11 = 550 cm3
Hencevolume of the given pentagonal pyramid = 550 cm3