Rectangular Pyramid Calculator
Calculate volume, surface area, and other properties of a rectangular pyramid
Calculation Results
Enter rectangular pyramid measurements to calculate all properties.
Base Area (B): square units
Height (h): units
Volume (V): cubic units
Lateral Area (L): square units
Surface Area (A): square units
Base Perimeter (P): units
Slant Height (length side): units
Slant Height (width side): units
Edge Length (apex to corner): units
Rectangular Pyramid Diagram
Rectangular Pyramid Formulas
Key Properties
- Base Area (B): Area of the rectangular base (l × w)
- Height (h): Perpendicular distance from base to apex
- Volume (V): Space contained within the pyramid
- Lateral Area (L): Area of the four triangular faces
- Surface Area (A): Total area including the base
- Base Perimeter (P): Perimeter of the rectangular base
- Slant Heights: Two different slant heights (one for each pair of triangular faces)
- Edge Length (e): Length from apex to base corner
Calculation Formulas
- Base Area (B): B = l × w
- Volume (V): V = ⅓ × l × w × h
- Surface Area (A): A = l × w + l × lₛ + w × wₛ
- Base Perimeter (P): P = 2 × (l + w)
- Slant Height (length side): lₛ = √(h² + (w/2)²)
- Slant Height (width side): wₛ = √(h² + (l/2)²)
- Lateral Area (L): L = l × lₛ + w × wₛ
- Edge Length (e): e = √(h² + (l/2)² + (w/2)²)
Where: l = base length, w = base width, h = height of pyramid, lₛ = slant height on length side, wₛ = slant height on width side
Example Calculation
For a rectangular pyramid with:
Base length (l) = 8 units, Base width (w) = 6 units, Height (h) = 10 units.Calculations:
- Base Area = 8 × 6 = 48 units²
- Base Perimeter = 2 × (8 + 6) = 28 units
- Volume = ⅓ × 8 × 6 × 10 = 160 units³
- Slant Height (length side) = √(10² + (6/2)²) ≈ 10.44 units
- Slant Height (width side) = √(10² + (8/2)²) ≈ 10.77 units
- Lateral Area ≈ (8 × 10.44) + (6 × 10.77) ≈ 83.52 + 64.62 ≈ 148.14 units²
- Surface Area ≈ 48 + 148.14 ≈ 196.14 units²
- Edge Length ≈ √(10² + (8/2)² + (6/2)²) ≈ √(100 + 16 + 9) ≈ 11.18 units
About Rectangular Pyramids
A rectangular pyramid is a three-dimensional shape with a rectangular base and four triangular faces meeting at a common apex. It has 5 faces, 8 edges, and 5 vertices. Unlike square pyramids, rectangular pyramids have two different slant heights due to the unequal side lengths.
Real-World Applications
- Architecture: Some pyramid structures use rectangular bases
- Packaging: Pyramid-shaped boxes with rectangular bases
- Design: Decorative elements and sculptures
- Mathematics: Study of polyhedrons and geometry
Types of Rectangular Pyramids
- Right Rectangular Pyramid: Apex is directly above the center of the base
- Oblique Rectangular Pyramid: Apex is not aligned with the base center
- Regular Rectangular Pyramid: All triangular faces are congruent isosceles triangles (only possible when base is square)
Rectangular Pyramid Components
Base Area (B)
B = l × w
Pyramid Height (h)
Lateral Area (L)
L = l × lₛ + w × wₛ
Surface Area (A)
A = l × w + l × lₛ + w × wₛ
Base Perimeter (P)
P = 2 × (l + w)
Volume (V)
V = ⅓ × l × w × h
How to Use the Rectangular Pyramid Calculator
The Rectangular Pyramid Calculator helps compute various properties based on input values. Here's how to use it:
1. Select Calculation Method
Choose what measurements you know:
- Base Dimensions & Height - Enter base length, width and pyramid height
- Base Area & Height - When you know the base area
- Volume & Base Area - To find the pyramid height
- Surface Area - When you know total surface area
- Slant Heights & Base - When you know both slant heights
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1: Base length = 8, Base width = 6, Height = 10
Example 2: Slant heights = 10.44 (length side) and 10.77 (width side), Base length = 8, Base width = 6
3. Get Results
The calculator will compute all properties:
- Base Area (B)
- Pyramid Height (h)
- Volume (V)
- Lateral Area (L)
- Surface Area (A)
- Base Perimeter (P)
- Slant Heights (lₛ and wₛ)
- Edge Length (e)
Practical Applications
- Calculate material needed for rectangular pyramid structures
- Determine paint required for pyramid-shaped objects
- Find storage capacity of pyramid-shaped containers
- Solve geometry problems involving rectangular pyramids
Rectangular Pyramid Formulas
| Parameter | Formula | Description |
|---|---|---|
| Base Length (l) | Given | Length of base |
| Base Width (w) | Given | Width of base |
| Base Area (B) | l × w | Area of the rectangular base |
| Height (h) | Given or calculated | Height from apex to base |
| Volume (V) | (l × w × h)/3 | Volume of the pyramid |
| Slant Height (length side) | √(h² + (w/2)²) | Height of triangular faces on length side |
| Slant Height (width side) | √(h² + (l/2)²) | Height of triangular faces on width side |
| Lateral Area (L) | l × lₛ + w × wₛ | Sum of areas of the 4 lateral faces |
| Surface Area (A) | l × w + l × lₛ + w × wₛ | Total surface area (base + lateral) |
| Base Perimeter (P) | 2 × (l + w) | Perimeter of the base |
| Edge Length (e) | √(h² + (l/2)² + (w/2)²) | Length from apex to base corner |
Frequently Asked Questions
A square pyramid has a square base (all sides equal) with four identical triangular faces, while a rectangular pyramid has a rectangular base (different length and width) with two pairs of different triangular faces.
A rectangular pyramid has two different slant heights - one for the triangular faces on the length side and another for the faces on the width side.
You can use either slant height formula to solve for height: h = √(lₛ² - (w/2)²) or h = √(wₛ² - (l/2)²) where lₛ is slant height on length side and wₛ is slant height on width side.