🔺 Right Rectangular Pyramid Calc: find A, V, A_l, A_b
Calculate all properties of a right rectangular pyramid
How to Use This Calculator
Enter Base Dimensions and Height
Input the length and width of the rectangular base, and the perpendicular height of the pyramid. Make sure all values are positive numbers.
Click Calculate
Press the "Calculate All Properties" button to compute the total surface area (A), volume (V), lateral area (A_l), and base area (A_b).
Review All Results
View all calculated properties: total surface area (all faces), volume (space inside), lateral area (four triangular sides), and base area (rectangular base).
Formulas
Base Area: A_b = l × w
Area of the rectangular base
Volume: V = (1/3) × A_b × h = (1/3)lwh
One-third the volume of a rectangular prism with same base and height
Lateral Area: A_l = sum of areas of 4 triangular faces
Two pairs of triangles with different slant heights
Total Surface Area: A = A_l + A_b
Lateral area plus base area
Where:
- l = length of the rectangular base
- w = width of the rectangular base
- h = perpendicular height from base to apex
- A = total surface area
- V = volume
- A_l = lateral surface area (four triangular faces)
- A_b = base area (rectangular base)
Note: The lateral area consists of 4 triangular faces. Two triangles have base l with slant height √(h² + (w/2)²), and two triangles have base w with slant height √(h² + (l/2)²).
Example 1: Pyramid with base 8 × 6 units, height 10 units
Base Area = 8 × 6 = 48 units²
Volume = (1/3) × 48 × 10 = 160 units³
Lateral Area ≈ 172.34 units² (calculated from triangular faces)
Total Surface Area = 172.34 + 48 = 220.34 units²
Example 2: Pyramid with base 10 × 8 units, height 12 units
Base Area = 10 × 8 = 80 units²
Volume = (1/3) × 80 × 12 = 320 units³
Lateral Area ≈ 245.86 units²
Total Surface Area = 245.86 + 80 = 325.86 units²
About Right Rectangular Pyramid Calculator
A right rectangular pyramid is a 3D shape with a rectangular base and four triangular faces meeting at an apex directly above the center of the base. This calculator finds all key properties: total surface area (A), volume (V), lateral area (A_l - the four triangular sides), and base area (A_b - the rectangular base).
When to Use This Calculator
- Architecture: Calculate volumes and surface areas for pyramid-shaped buildings or structures
- Engineering: Design pyramid-shaped components with specific volume or area requirements
- Mathematics Education: Teach students about 3D geometry and pyramid properties
- Construction: Estimate materials (concrete, paint) for pyramid structures
- Design: Plan pyramid shapes with specific volume or surface area specifications
- Packaging: Calculate capacity and surface area for pyramid-shaped containers
Why Use Our Calculator?
- ✅ Complete Properties: Calculates A, V, A_l, and A_b all at once
- ✅ Instant Results: Get all measurements immediately from base dimensions and height
- ✅ Step-by-Step Display: See formulas and calculations for each property
- ✅ 100% Accurate: Uses precise mathematical formulas
- ✅ Educational: Helps understand pyramid geometry relationships
- ✅ Completely Free: No registration required
Understanding Right Rectangular Pyramids
A right rectangular pyramid consists of:
- Rectangular Base: A rectangle with length l and width w, area = l × w
- Four Triangular Faces: Four triangles connecting the base edges to the apex
- Apex: The point at the top, directly above the center of the base
- Height (h): Perpendicular distance from base to apex
- Volume: One-third the volume of a rectangular prism with same base and height = (1/3)lwh
- Lateral Area: Sum of areas of the four triangular side faces
- Base Area: Area of the rectangular base = l × w
- Total Surface Area: Lateral area + base area
Real-World Applications
Architecture: A pyramid roof has rectangular base 10 m × 8 m and height 6 m. Base area = 80 m², Volume = (1/3) × 80 × 6 = 160 m³ (airspace). Lateral area ≈ 147.29 m² (roofing material for sides). Total surface area ≈ 227.29 m² (all surfaces).
Construction: A pyramid structure with base 5 m × 4 m and height 3 m has volume = (1/3) × 20 × 3 = 20 m³ (concrete needed) and total surface area ≈ 45.84 m² (all surfaces for finishing).
Packaging: A pyramid gift box with base 15 cm × 12 cm and height 10 cm has volume = (1/3) × 180 × 10 = 600 cm³ (capacity) and base area = 180 cm² (bottom surface).
Frequently Asked Questions
What is a right rectangular pyramid?
A right rectangular pyramid is a 3D shape with a rectangular base and four triangular faces meeting at an apex that is directly above the center of the base. "Right" means the apex is perpendicular to the base center, not slanted.
How is this different from a square pyramid?
A square pyramid has a square base (length = width), while a rectangular pyramid has a rectangular base (length ≠width). If length = width, a rectangular pyramid becomes a square pyramid. The formulas are similar, but square pyramids have more symmetry.
Why is the volume formula (1/3)lwh?
A pyramid has exactly one-third the volume of a prism with the same base and height. This fundamental relationship applies to all pyramids. For a rectangular pyramid: V = (1/3) × (base area) × height = (1/3)lwh.
How is lateral area calculated?
Lateral area is the sum of areas of the four triangular faces. Two triangles have base l with slant height √(h² + (w/2)²), and two triangles have base w with slant height √(h² + (l/2)²). Each triangle area = (1/2) × base × slant height.
Can I use this for an oblique pyramid?
The height (h) must be the perpendicular distance from the base to the apex. For an oblique pyramid, use the vertical height, not the slant height. The volume formula still works, but lateral area calculation becomes more complex.
What if the base is a square?
If length = width, you have a square pyramid, which is a special case of a rectangular pyramid. Enter the same value for both length and width. The formulas still work, and you'll get the same results as a dedicated square pyramid calculator.