Hexagon Calculator

Introduction to Hexagons

A hexagon is a six-sided polygon that has played a significant role throughout mathematics, nature, and human design. The word "hexagon" comes from the Greek words "hex" meaning six and "gonia" meaning angle or corner. Beyond its simple definition, hexagons reveal remarkable mathematical properties and natural efficiency.

Types of Hexagons

Mathematical Properties

• The sum of interior angles in any hexagon is 720° (calculated using the formula (n-2) × 180° where n=6)
• A regular hexagon can be divided into six equilateral triangles
• The area of a regular hexagon with side length s is (3√3 × s²) / 2
• The perimeter is simply 6 × s
• The apothem (distance from center to midpoint of any side) equals (√3 × s) / 2
• Regular hexagons can tessellate (tile a plane with no gaps or overlaps)

Hexagons in Nature

Hexagons appear naturally in many forms due to their efficiency and structural advantages:

Hexagons in Human Design

The efficiency and aesthetic appeal of hexagons have made them popular in various fields:

• Architecture: Hexagonal patterns in buildings, tiling, and structures provide both stability and visual interest
• Engineering: Hexagonal patterns maximize strength while minimizing material in structures like geodesic domes
• Gaming: Many board games use hexagonal grids because they allow movement in six directions (compared to four in square grids)
• Technology: Carbon nanotubes and graphene have hexagonal molecular structures that provide exceptional strength
• Design: Hexagons are popular in modern design for their clean, geometric aesthetic

Hexagonal Tessellation

Regular hexagons are one of only three regular polygons (along with triangles and squares) that can tessellate a plane. This property makes hexagonal tiling extremely useful in applications ranging from floor tiles to cellular automata in computer science.

The Hexagonal Close Packing

When circles are arranged in the most efficient way on a plane, their centers form a hexagonal pattern. This hexagonal close packing achieves approximately 90.69% efficiency in covering the plane, making it the most efficient arrangement possible.

Hexagons in Mathematics

Beyond basic geometry, hexagons appear in various mathematical contexts:

• Pascal's Triangle: The hexagonal numbers appear in Pascal's Triangle
• Graph Theory: Regular hexagonal tilings represent graphs with interesting properties
• Hexagonal Number System: Base-6 or heximal numeral system (distinct from hexadecimal)

A regular hexagon is a six-sided polygon where all sides are equal in length and all angles are equal. It's a common shape found in nature, such as in honeycombs, and is used in various applications from architecture to engineering.