Volume of a Pyramid Formula

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 base, and the volume is measured in cubic units such as cm³, m³, or in³.

Relation Between Slant Height and Height

In a pyramid, the 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 prism, it 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 polygon, so 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

Hence, volume 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 cm, respectively?

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

Hence, volume 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 cm, respectively, and 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

Hence, volume 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 cm, respectively, and 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

Hence, volume 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 cm

Area of hexagonal base:
A = (3√3 / 2) × a²
A = (3√3 / 2) × (8)²
A = (3√3 / 2) × 64
A = 96√3 cm²

Volume 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

Hence, volume of the given pentagonal pyramid = 550 cm³