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
Width of the rectangular base
Perpendicular height from base to apex
Calculation Results
Volume Formula: V = (a × b × h) / 3
Surface Area Formula: A = ab + a√((b/2)² + h²) + b√((a/2)² + h²)
Input Validation
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
Faces
1 rectangular base + 4 triangular faces
Vertices
4 at base corners + 1 at apex
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.