Rectangular Pyramid Volume Calculator

Calculate volume and surface area of a rectangular pyramid with step-by-step solutions

Calculate Rectangular Pyramid Volume

Length of the rectangular base

cm

Width of the rectangular base

cm

Perpendicular height from base to apex

Calculation Results

0
Volume (cm³)
0
Surface Area (cm²)
0
Base Area (cm²)
0
0
Slant Heights (cm)

Volume Formula: V = (a × b × h) / 3

Surface Area Formula: A = ab + a√((b/2)² + h²) + b√((a/2)² + h²)

Input Validation

⚠️ Please enter a positive base length value.
⚠️ Please enter a positive base width value.
⚠️ Please enter a positive height value.

Example Calculation

Egyptian Pyramid Model

Base dimensions: Length = 7 cm, Width = 5 cm

Height: 10 cm

Base area: 7 × 5 = 35 cm²

Step-by-Step Solution

1. Volume = (a × b × h) / 3

2. Volume = (7 × 5 × 10) / 3

3. Volume = 350 / 3

4. Volume = 116.67 cm³

Pyramid Properties

5

Faces

1 rectangular base + 4 triangular faces

5

Vertices

4 at base corners + 1 at apex

8

Edges

4 base edges + 4 lateral edges

Formula Guide

Volume

V = (a × b × h) / 3

Where a, b are base dimensions and h is height

Surface Area

A = ab + a√((b/2)² + h²) + b√((a/2)² + h²)

Base area + lateral face areas

Slant Height

s = √((d/2)² + h²)

Where d is the perpendicular base dimension

Understanding Rectangular Pyramid Volume

What is a Rectangular Pyramid?

A rectangular pyramid is a three-dimensional geometric shape with a rectangular base and four triangular faces that meet at a single point called the apex. The height is the perpendicular distance from the base to the apex.

Key Components

  • Base: A rectangle with length (a) and width (b)
  • Height (H): Perpendicular distance from base to apex
  • Slant Height: Distance along a face from base edge to apex
  • Apex: The pointed top where all triangular faces meet

Volume Formula Derivation

V = (Base Area × Height) / 3

V = (a × b × h) / 3

The volume formula comes from the general pyramid volume formula where the volume equals one-third of the base area times the height. This relationship holds for any pyramid regardless of the base shape.

Real-World Applications

  • Architecture and construction planning
  • Packaging and container design
  • Material quantity calculations
  • Engineering and CAD modeling

Mathematical Properties

Volume Relationship

The volume is always 1/3 of the corresponding rectangular prism (base area × height).

Scaling Properties

If all dimensions are scaled by factor k, the volume scales by k³ and surface area by k².

Euler's Formula

For any pyramid: V - E + F = 2, where V=5 vertices, E=8 edges, F=5 faces.