Hexagonal Prism Calculator

Online calculator and formulas for calculating a hexagonal prism

The Six-Sided Prism - Perfect Hexagonal Symmetry in 3D!

Hexagonal Prism Calculator

The Hexagonal Prism

A hexagonal prism is a prismatic solid with hexagonal bases and rectangular sides. Perfect six-fold symmetry.

Enter parameters
Length of hexagon side
What to calculate?
Value of known parameter
Hexagonal Prism Calculation Results
Height:
Surface:
Base area:
Hexagonal Prism Properties

The six-sided prism: Two parallel hexagonal bases connected by rectangular faces

2 Hexagonal bases 6 Rectangular faces 18 Edges 12 Vertices

Hexagonal Prism Structure

Hexagonal Prism

The prism with hexagonal cross-section.
Perfect six-fold rotational symmetry.

What is a Hexagonal Prism?

A hexagonal prism is a fundamental prismatic shape with remarkable properties:

  • Definition: Prism with hexagonal cross-section
  • Bases: Two parallel regular hexagons
  • Sides: Six rectangular faces
  • Vertices: 12 vertices total
  • Edges: 18 edges (6+6+6)
  • Symmetry: Six-fold rotational symmetry

Geometric Properties of the Hexagonal Prism

The hexagonal prism displays beautiful geometric characteristics:

Hexagonal Parameters
  • Bases: 2 regular hexagons
  • Side faces: 6 rectangles
  • Internal angles: 120° in hexagon
  • Euler formula: V - E + F = 12 - 18 + 8 = 2
Special Properties
  • Tessellation: Hexagons tile the plane perfectly
  • Natural occurrence: Found in honeycombs
  • Efficiency: Maximum area for given perimeter
  • Six-fold symmetry: Rotational and reflection symmetry

Mathematical Relationships

The hexagonal prism follows elegant mathematical principles:

Square Root Relationships
√3 factor in area

Hexagonal areas involve √3. Elegant and precise formulas.

Practical Calculations
Simple formulas

Clear relationships between dimensions. Easy to compute and understand.

Applications of the Hexagonal Prism

Hexagonal prisms are found everywhere in nature and technology:

Nature & Biology
  • Honeycomb structures
  • Crystal formations
  • Basalt columns
  • Snowflake patterns
Engineering & Manufacturing
  • Hex bolts and nuts
  • Structural beams
  • Pencil cross-sections
  • Efficient packing
Architecture & Construction
  • Hexagonal tiles
  • Dome structures
  • Modular construction
  • Space frames
Games & Design
  • Board game tiles
  • Game grids
  • Decorative patterns
  • Artistic designs

Formulas for the regular hexagonal prism

Base area (A)
\[A = 1.5 \cdot \sqrt{3} \cdot a^2\] \[A = \frac{3\sqrt{3}}{2} \cdot a^2\]

Area of the regular hexagonal base

Surface area (S)
\[S = 3 \cdot \sqrt{3} \cdot a^2 + 6 \cdot a \cdot h\]

Total surface area: 2 hexagons + 6 rectangles

Volume (V)
\[V = 1.5 \cdot \sqrt{3} \cdot a^2 \cdot h\] \[V = \frac{3\sqrt{3}}{2} \cdot a^2 \cdot h\]

Volume equals base area times height

Height (h)
\[h = \frac{2V}{3 \sqrt{3} a^2}\]

Height when volume and side length are known

Hexagon properties
Interior angle
120°
Perimeter
6a
Diagonal
2a
Apothem
a√3/2

Key properties of the regular hexagon base

Example calculation for a Hexagonal Prism

Given
Side length a = 5 Volume V = 140

Calculate: Height and surface area of the hexagonal prism

1. Base area calculation
\[A = 1.5 \times \sqrt{3} \times 5^2\] \[A = 1.5 \times 1.732 \times 25 = 64.95\]

The base area is 64.95 square units

2. Height calculation
\[h = \frac{2 \times 140}{3 \times \sqrt{3} \times 25}\] \[h = \frac{280}{129.9} = 2.16\]

The height is 2.16 units

3. Surface area calculation
\[S = 3 \times \sqrt{3} \times 25 + 6 \times 5 \times 2.16\] \[S = 129.9 + 64.8 = 194.7\]

The surface area is 194.7 square units

4. Verification
\[V = 64.95 \times 2.16 = 140.3\]

✓ Close to given volume of 140

Calculation verified successfully

Summary
Side: 5.0
Height: 2.16
Base area: 64.95
Surface: 194.7

Complete analysis of the hexagonal prism with given parameters

The Hexagonal Prism: Nature's Perfect Shape

The hexagonal prism represents one of nature's most efficient and beautiful geometric forms. From the hexagonal cells in honeycomb structures to the basalt columns of Giant's Causeway, this shape demonstrates the perfect balance between structural strength and material efficiency. Its six-fold symmetry creates patterns that tile perfectly, making it an ideal choice for both natural formations and human engineering applications.

The Geometry of Efficiency

The hexagonal prism showcases mathematical and practical perfection:

  • Perfect tessellation: Hexagons tile the plane without gaps or overlaps
  • Maximum efficiency: Largest area for given perimeter among regular polygons
  • Six-fold symmetry: Rotational symmetry every 60 degrees
  • Natural occurrence: Found throughout the natural world
  • Structural strength: Distributes stress efficiently
  • Simple mathematics: Elegant formulas involving √3
  • Practical applications: Used extensively in engineering and design

Mathematical Beauty and Natural Wisdom

Honeycomb Efficiency

Bees have discovered that hexagonal cells require the least amount of wax for maximum storage space, demonstrating nature's optimization of the hexagonal form.

Crystal Structures

Many crystals naturally form hexagonal prisms due to their atomic structure, showing how geometry emerges from molecular arrangements.

Engineering Applications

From hex bolts that resist turning to structural beams that provide strength, engineers have adopted the hexagon for its superior mechanical properties.

Mathematical Elegance

The hexagonal prism's formulas involve √3, connecting it to the fundamental relationships in geometry and trigonometry.

Conclusion

The hexagonal prism stands as a testament to the unity of mathematical beauty and practical efficiency. Its appearance throughout nature - from snowflakes to basalt formations - demonstrates that certain geometric forms represent optimal solutions to physical constraints. In human applications, from ancient architecture to modern manufacturing, the hexagonal prism continues to prove its worth as both a structural element and an aesthetic choice. For students of geometry, it provides an excellent example of how mathematical principles manifest in the real world, bridging the gap between abstract theory and tangible applications. The hexagonal prism truly embodies the perfect marriage of form and function, efficiency and elegance.