Rectangular Pyramid Volume Formula and Calculator

Understanding the Rectangular Pyramid

A rectangular pyramid is a polyhedron featuring a rectangular base and triangular lateral faces that converge at a single point called the apex. Iconic real-world examples include the Egyptian Pyramids and the Louvre Pyramid in Paris.

Key components of a right rectangular pyramid are the length (a) and width (b) of the base, and the perpendicular height (H) from the base's center to the apex.

Essential Formulas for Volume and Surface Area

To determine the volume or capacity of a rectangular pyramid, apply the following mathematical formula:

V = (a * b * H) / 3

• a: Length of the rectangular base.
• b: Width of the rectangular base.
• H: Perpendicular height of the pyramid from the base to the apex.

For calculating the total surface area (including the base and all triangular faces), use this formula:

A = a*b + a√((b/2)² + H²) + b√((a/2)² + H²)

Practical Calculation Example

Let's calculate the volume and surface area of a pyramid with a base length (a) of 7 cm, a base width (b) of 5 cm, and a height (H) of 10 cm.

Step 1: Calculate the Base Area

Base Area = a * b = 7 cm * 5 cm = 35 cm²

Step 2: Calculate the Volume

V = (Base Area * H) / 3 = (35 cm² * 10 cm) / 3 ≈ 116.67 cm³

The volume of this rectangular pyramid is approximately 116.67 cubic centimeters.

Step 3: Calculate the Surface Area

Using the surface area formula:

A = (7*5) + 7√((5/2)² + 10²) + 5√((7/2)² + 10²)
A ≈ 35 + 7√(6.25 + 100) + 5√(12.25 + 100)
A ≈ 35 + 7√106.25 + 5√112.25
A ≈ 35 + (7*10.308) + (5*10.595)
A ≈ 35 + 72.156 + 52.975 ≈ 160.13 cm²

The total surface area is approximately 160.13 square centimeters.