Pentagonal Prism Calculator
Calculate volume, surface area, and other properties of a regular pentagonal prism
Calculation Results
Enter pentagonal prism measurements to calculate all properties.
Side Length (a): units
Base Area (B): square units
Height (h): units
Volume (V): cubic units
Lateral Area (L): square units
Surface Area (S): square units
Base Perimeter (P): units
Pentagonal Prism Diagram
Pentagonal Prism Formulas
Key Properties
- Side Length (a): Length of one side of the pentagonal base
- Base Area (B): Area of the pentagonal base
- Height (h): Distance between the two pentagonal bases
- Volume (V): Space contained within the prism
- Lateral Area (L): Area of the five rectangular faces
- Surface Area (S): Total area including both pentagonal bases
- Base Perimeter (P): Perimeter of the pentagonal base
Calculation Formulas
- Base Area (B): B = (5/4)a²cot(π/5) ≈ 1.72048a²
- Volume (V): V = B × h
- Lateral Area (L): L = 5a × h
- Surface Area (S): S = L + 2B
- Base Perimeter (P): P = 5a
Where: a = side length of pentagon, h = height of prism
Example Calculation
For a pentagonal prism with:
Side length (a) = 4 units, Prism height (h) = 10 units.Calculations:
- Base Area ≈ 1.72048 × 4² ≈ 27.528 units²
- Base Perimeter = 5 × 4 = 20 units
- Volume ≈ 27.528 × 10 ≈ 275.28 units³
- Lateral Area = 5 × 4 × 10 = 200 units²
- Surface Area ≈ 200 + (2 × 27.528) ≈ 255.06 units²
About Pentagonal Prisms
A pentagonal prism is a three-dimensional shape with two identical pentagonal bases connected by five rectangular lateral faces. It has 7 faces, 15 edges, and 10 vertices. When the pentagon is regular (all sides and angles equal), the prism is called a regular pentagonal prism.
Real-World Applications
- Architecture: Some unique buildings use pentagonal prism shapes
- Optics: Pentagonal prisms are used in cameras and optical instruments
- Design: Decorative elements and sculptures often use this shape
- Packaging: Some specialty packages use pentagonal prism shapes
Special Cases
- Regular Pentagonal Prism: All sides of the base are equal and all angles are equal
- Right Pentagonal Prism: Lateral faces are rectangles (as opposed to parallelograms)
- Uniform Pentagonal Prism: All edges are equal length (becomes a pentagonal trapezohedron)
Pentagonal Prism Components
Base Area (B)
B ≈ 1.72048a²
Prism Height (h)
Lateral Area (L)
L = 5ah
Surface Area (S)
S = L + 2B
Base Perimeter (P)
P = 5a
Volume (V)
V = B × h
How to Use the Pentagonal Prism Calculator
The Pentagonal Prism Calculator helps compute various properties based on input values. Here's how to use it:
1. Select Calculation Method
Choose what measurements you know:
- Side Length & Height - Enter the pentagon side and prism height
- Base Area & Height - When you know the base area
- Volume & Height - To find the side length
- Lateral Area & Height - To find the side length
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1: Side Length = 4, Prism Height = 10
Example 2: Base Area = 27.528, Prism Height = 10
3. Get Results
The calculator will compute all properties:
- Side Length (a)
- Base Area (B)
- Prism Height (h)
- Volume (V)
- Lateral Area (L)
- Surface Area (S)
- Base Perimeter (P)
Practical Applications
- Calculate material needed for pentagonal structures
- Determine paint required for pentagonal prism-shaped objects
- Find storage capacity of pentagonal containers
- Solve geometry problems involving pentagonal prisms
Frequently Asked Questions
A pentagonal prism has two pentagonal bases and five rectangular faces, while a pentagonal pyramid has one pentagonal base and five triangular faces meeting at a point (apex).
A pentagonal prism has 7 faces - two pentagonal bases and five rectangular lateral faces.
Use the formula: a = √(V / (1.72048 × h)) where V is volume and h is height.