Pentagonal Pyramid Calculator
Calculate volume, surface area, and other properties of a pentagonal pyramid
Calculation Results
Enter pentagonal 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 (l): units
Apothem (ap): units
Pentagonal Pyramid Diagram
Pentagonal Pyramid Formulas
Key Properties
- Base Area (B): Area of the pentagonal base
- Height (h): Perpendicular distance from base to apex
- Volume (V): Space contained within the pyramid
- Lateral Area (L): Area of the five triangular faces
- Surface Area (A): Total area including the base
- Base Perimeter (P): Perimeter of the pentagonal base
- Slant Height (l): Height of the triangular faces
- Apothem (ap): Distance from center to midpoint of a base side
Calculation Formulas
- Base Area (Regular): B = (5/4) × a² × cot(π/5)
- Base Area (General): B = (5/2) × a × ap
- Volume (V): V = ⅓ × B × h
- Surface Area (A): A = B + L
- Base Perimeter (P): P = 5 × a
- Regular Pentagon Apothem: ap = a / (2 × tan(π/5))
- Slant Height (Regular): l = √(h² + ap²)
- Lateral Area (Regular): L = (5/2) × a × l
Where: a = base side length, h = height of pyramid, ap = apothem of base, l = slant height
Example Calculation
For a regular pentagonal pyramid with:
Edge length (a) = 6 units, Height (h) = 8 units.Calculations:
- Apothem ≈ 6 / (2 × tan(36°)) ≈ 4.13 units
- Base Area ≈ (5/4) × 6² × cot(36°) ≈ 61.94 units²
- Base Perimeter = 5 × 6 = 30 units
- Volume ≈ ⅓ × 61.94 × 8 ≈ 165.17 units³
- Slant Height ≈ √(8² + 4.13²) ≈ 8.99 units
- Lateral Area ≈ (5/2) × 6 × 8.99 ≈ 134.85 units²
- Surface Area ≈ 61.94 + 134.85 ≈ 196.79 units²
About Pentagonal Pyramids
A pentagonal pyramid is a three-dimensional shape with a pentagonal base and five triangular faces meeting at a common apex. It has 6 faces, 10 edges, and 6 vertices. When the base is a regular pentagon and the apex is directly above the center, it's called a regular pentagonal pyramid.
Real-World Applications
- Architecture: Some decorative roof elements and spires
- Design: Unique geometric sculptures and art pieces
- Chemistry: Molecular structures with pentagonal symmetry
- Mathematics: Study of polyhedrons and geometry
Types of Pentagonal Pyramids
- Regular Pentagonal Pyramid: Base is a regular pentagon, apex directly above center
- Irregular Pentagonal Pyramid: Base is an irregular pentagon
- Right Pentagonal Pyramid: Apex is directly above the base's centroid
- Oblique Pentagonal Pyramid: Apex is not aligned with the base's centroid
Pentagonal Pyramid Components
Base Area (B)
B = (5/2) × a × ap
Pyramid Height (h)
Lateral Area (L)
L = (5/2) × a × l
Surface Area (A)
A = B + L
Base Perimeter (P)
P = 5 × a
Volume (V)
V = ⅓ × B × h
How to Use the Pentagonal Pyramid Calculator
The Pentagonal 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 side length, apothem 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
- Regular Pentagon - When base is a regular pentagon
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1: Side length = 6, Apothem = 4.13, Height = 8
Example 2: Edge length = 6, Height = 8 (for regular pentagon)
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 Height (l)
- Apothem (ap)
Practical Applications
- Calculate material needed for pentagonal pyramid structures
- Determine paint required for pyramid-shaped objects
- Find storage capacity of pentagonal pyramid containers
- Solve geometry problems involving pentagonal pyramids
Pentagonal Pyramid Formulas
| Parameter | Formula | Description |
|---|---|---|
| Base Side Length (a) | Given | Length of pentagon's side |
| Base Area (Regular) | (5/4) × a² × cot(π/5) | Area of regular pentagonal base |
| Base Area (General) | (5/2) × a × ap | Area using side length and apothem |
| Height (h) | Given or calculated | Height from apex to base |
| Volume (V) | (B × h)/3 | Volume of the pyramid |
| Apothem (Regular) | a / (2 × tan(π/5)) | Distance from center to side midpoint |
| Slant Height (Regular) | √(h² + ap²) | Height of triangular faces |
| Lateral Area (Regular) | (5/2) × a × l | Sum of areas of the 5 lateral faces |
| Surface Area (A) | B + L | Total surface area (base + lateral) |
| Base Perimeter (P) | 5 × a | Perimeter of the base |
Frequently Asked Questions
A pentagonal pyramid has a pentagonal base and triangular faces meeting at an apex, while a pentagonal prism has two pentagonal bases connected by rectangular faces.
A pentagonal pyramid has 6 faces - one pentagonal base and five triangular lateral faces.
Use the formula: l = √(h² + ap²) where h is height and ap is apothem.