Surface Area of a Rectangular Pyramid Calculator
Calculate total surface area, lateral area, and volume of rectangular pyramids with step-by-step solutions
Calculate Rectangular Pyramid Properties
Length of rectangular base
Width of rectangular base
Perpendicular height from apex to base
Rectangular Pyramid Formulas
Base Area: A_b = a × b
Slant Heights: l_a = √(H² + (b/2)²), l_b = √(H² + (a/2)²)
Lateral Area: A_l = a × l_a + b × l_b
Total Surface Area: A_t = A_b + A_l
Volume: V = (1/3) × A_b × H
Example Calculations
Example 1: From Competitor
Given: length = 5 cm, width = 4 cm, height = 5 cm
Base area: 5 × 4 = 20 cm²
Slant heights: l_a = 5.39 cm, l_b = 5.83 cm
Lateral area: 49.29 cm²
Total area: 20 + 49.29 = 69.29 cm²
Example 2: Square Base
Given: length = 6 m, width = 6 m, height = 8 m
Base area: 6 × 6 = 36 m²
Slant height: √(8² + 3²) = 8.54 m
Lateral area: 4 × (6 × 8.54) = 205.0 m²
Total area: 36 + 205.0 = 241.0 m²
Quick Reference
Key Formulas
A_t = a×b + a×l_a + b×l_b
l_a = √(H² + (b/2)²)
l_b = √(H² + (a/2)²)
V = (1/3) × a × b × H
Components
1 rectangular base
4 triangular faces
5 vertices, 8 edges
2 pairs of congruent faces
Rectangular Pyramid Facts
Quick Tips
Right pyramid has apex directly above base center
Two different slant heights for rectangular base
Lateral area includes all four triangular faces
Height is perpendicular distance to base
Understanding Rectangular Pyramid Surface Area
What is a Rectangular Pyramid?
A rectangular pyramid is a 3D shape with a rectangular base and four triangular faces that meet at a single point (apex). Unlike square pyramids, it has two different slant heights due to the rectangular base having different length and width.
Surface Area Components
The total surface area consists of the rectangular base area plus the areas of four triangular faces. Since opposite faces are congruent, we calculate two different face areas and double them.
Formula Derivation
Step 1: Base Area
A_b = a × b (rectangular area)
Step 2: Slant Heights
l_a = √(H² + (b/2)²), l_b = √(H² + (a/2)²)
Step 3: Lateral Area
A_l = a×l_a + b×l_b (four triangular faces)
Real-World Applications
Architecture:
Egyptian and Mayan pyramids, modern architectural elements
Engineering:
Vickers hardness testing uses pyramid indenter for material testing
Manufacturing:
Calculate material needed for pyramid-shaped containers and structures