Hexagon Calculator
Calculate area, perimeter, and both types of diagonals for regular hexagons
Input Values
Results
Step-by-Step Solution
A = (3√3/2) × s²
A = (3√3/2) × 10²
A = 2.598076 × 100.0000
A = 0.0000
Short diagonal: d₁ = s√3 = 10 × √3 = 0.0000
Long diagonal: d₂ = 2s = 2 × 10 = 0.0000
a = s√3/2
a = 0.0000 × √3/2
a = 0.0000
For a regular hexagon, radius = side length
r = s = 0.0000
Understanding Hexagons
What is a Hexagon?
A hexagon is a six-sided polygon. A regular hexagon has all six sides of equal length and all six interior angles equal to 120°. Regular hexagons are one of only three regular polygons that can tessellate (tile) a plane, along with equilateral triangles and squares.
Special Properties of Regular Hexagons
- Radius equals side length: Unlike most polygons, r = s for a regular hexagon
- Can be divided into 6 equilateral triangles: Each with side length s
- Two types of diagonals: Short (s√3) and long (2s)
- Perfect tessellation: No gaps when tiling a plane
- Efficient packing: Maximizes area for given perimeter in tessellation
Hexagon Formulas
Area: A = (3√3/2) × s² ≈ 2.598s²
Perimeter: P = 6s
Apothem: a = s√3/2 ≈ 0.866s
Short Diagonal: d₁ = s√3 ≈ 1.732s
Long Diagonal: d₂ = 2s
Radius: r = s (unique to hexagons!)
Interior Angle: 120°
Exterior Angle: 60°
Hexagons in Nature and Design
- Honeycomb: Bees build hexagonal cells for optimal space and material efficiency
- Snowflakes: Exhibit six-fold symmetry due to water molecule structure
- Carbon structures: Graphene and benzene rings have hexagonal arrangements
- Basalt columns: Form hexagonal shapes when lava cools (Giant's Causeway)
- Hardware: Nuts and bolt heads use hexagonal shapes for better grip
Frequently Asked Questions
Why do bees use hexagons in honeycombs?
Hexagons are the most efficient shape for storing honey while using the least amount of wax. They tessellate perfectly with no gaps, and among all tessellating shapes, hexagons provide the maximum area for a given perimeter. This is known as the honeycomb conjecture, proven mathematically in 1999.
What are the two types of diagonals in a hexagon?
A regular hexagon has 9 diagonals total, but they come in two lengths. Short diagonals (length s√3) connect vertices separated by one vertex. Long diagonals (length 2s) connect opposite vertices and pass through the center. There are 6 short diagonals and 3 long diagonals.
Why is the radius equal to the side length?
This unique property occurs because a regular hexagon can be divided into 6 equilateral triangles from the center. In an equilateral triangle, all sides are equal, so the radius (from center to vertex) equals the side length of the hexagon.
Can hexagons tessellate?
Yes, regular hexagons perfectly tessellate a plane. Their 120° interior angles allow exactly three hexagons to meet at each vertex (3 × 120° = 360°). This is why hexagonal tilings are common in floor tiles, game boards, and natural structures.
How do you construct a regular hexagon?
Draw a circle and mark a point on it. Without changing your compass width, place the compass point on that mark and make another mark on the circle. Repeat 6 times total. Connect the 6 marks to form a perfect regular hexagon. This works because the radius equals the side length.
What is the apothem of a hexagon?
The apothem is the perpendicular distance from the center to the midpoint of any side. For a regular hexagon, a = s√3/2. It's also the radius of the largest circle that fits inside the hexagon (the incircle).
How many lines of symmetry does a hexagon have?
A regular hexagon has 6 lines of symmetry: 3 lines connecting opposite vertices (through the center) and 3 lines connecting midpoints of opposite sides. It also has rotational symmetry of order 6, meaning it looks the same after rotation by 60°.
What is the area formula derivation?
A regular hexagon can be divided into 6 equilateral triangles, each with side s and area (s²√3)/4. Total area = 6 × (s²√3)/4 = (6s²√3)/4 = (3s²√3)/2. This elegant formula comes directly from the hexagon's geometric structure.