Hexagon Calculator — Area & Perimeter
Calculate the area, perimeter, and diagonals of a regular hexagon instantly.
Formula
Area = (3√3/2)s²
The area is calculated by summing the areas of 6 equilateral triangles with side length 's'.
Worked Examples
Example 1: All Properties from a Side of 10
Problem:A regular hexagon has a side length of 10 cm. Find its area, perimeter, apothem, long diagonal and interior angle.
Solution:Area = (3√3/2)s² = 2.598076 × 100 = 259.81 cm² Perimeter = 6s = 60 cm Apothem (centre to the middle of a side) = (√3/2)s = 0.866025 × 10 = 8.660 cm Circumradius (centre to a corner) = s = 10 cm, because a regular hexagon splits into six equilateral triangles Long diagonal = 2s = 20 cm Interior angle = (6-2) × 180 / 6 = 120 degrees Cross-check with the polygon formula A = ½ × perimeter × apothem = 0.5 × 60 × 8.660 = 259.81 cm². Matches.
Result:Area 259.81 cm² | Perimeter 60 cm | Apothem 8.660 cm | Diagonal 20 cm
Example 2: Smaller Hexagon and the Square-Law Scaling
Problem:Find the area and perimeter of a regular hexagon with side 5 cm, and compare it with the side-10 hexagon above.
Solution:Perimeter = 6 × 5 = 30 cm Area = (3√3/2) × 25 = 2.598076 × 25 = 64.95 cm² Apothem = 0.866025 × 5 = 4.330 cm Halving the side halves the perimeter (60 to 30) but quarters the area (259.81 to 64.95), because area scales with the square of the length. That square law is why a honeycomb built from short cells uses far less wax per unit of stored volume than one built from long cells.
Result:Area 64.95 cm² | Perimeter 30 cm | Apothem 4.330 cm
Example 3: Working Backwards from a Known Area
Problem:A hexagonal patio must cover 50 square metres. What side length is needed?
Solution:Rearrange A = (3√3/2)s² to s = the square root of (2A / (3√3)). 3√3 = 5.196152 s² = 2(50) / 5.196152 = 100 / 5.196152 = 19.2450 s = 4.3869 m Check: (3√3/2) × 19.2450 = 2.598076 × 19.2450 = 50.00 m². Correct. Perimeter = 6 × 4.3869 = 26.32 m of edging required.
Result:Side = 4.387 m | Perimeter = 26.32 m of edging
Frequently Asked Questions
What is a regular hexagon?
A regular hexagon is a six-sided polygon where all sides are equal in length and all interior angles are equal (120 degrees).
How do you calculate the area of a hexagon?
The formula is Area = (3√3 / 2) × side². This is derived by dividing the hexagon into 6 equilateral triangles.
What is the perimeter formula?
Perimeter = 6 × side length.
Why do bees build hexagonal honeycombs?
Hexagons are the most efficient shape to tile a flat surface (tessellation). They minimize the amount of wax needed to create a given volume of storage space (The Honeycomb Conjecture).
What is the "Long Diagonal"?
The distance between opposite vertices, passing through the center. It is equal to 2 × side length.
What is the "Short Diagonal"?
The distance between parallel flat sides. It is equal to √3 × side length (approx 1.732 × s).
What is the sum of interior angles?
For any polygon, sum = (n-2) × 180°. For a hexagon (n=6), sum = 4 × 180° = 720°. Each angle = 720° / 6 = 120°.
Where else do we see hexagons?
Snowflakes, basalt columns (Giant's Causeway), insect eyes, and carbon molecules (graphene/benzene) often form hexagonal structures.
Is a hexagon a strong shape?
Yes, because it efficiently distributes force. It is widely used in engineering and materials science (honeycomb structures in aerospace).
What is the difference between perimeter and area?
Perimeter measures the total boundary length of a 2D shape and is expressed in linear units (meters, feet). Area measures the space enclosed within that boundary and is expressed in square units (m², ft²). For a rectangle, perimeter = 2(l + w) and area = l × w. For a circle, circumference (perimeter) = 2πr and area = πr². Importantly, two shapes can have identical perimeters but very different areas — a 1×9 rectangle and a 3×3 square both have perimeter 20, but areas of 9 and 9 respectively only when both are square; a 4×6 rectangle has area 24 while a 1×9 has just 9. This distinction matters in fencing (perimeter) versus flooring or painting (area) problems.
Background & Theory
Regular Hexagon Properties:
- Sides: 6 equal sides
- Angles: 6 equal interior angles of 120°
- Tessellation: Can cover a plane without gaps (like squares and triangles).
Formulas:
- Area: $A = \\frac{3\\sqrt{3}}{2} s^2 \\approx 2.598 s^2$
- Perimeter: $P = 6s$
- Long Diagonal (d): $d = 2s$
- Short Diagonal (h): $h = \\sqrt{3}s$
- Inradius (apothem): $r = \\frac{\\sqrt{3}}{2}s$
- Circumradius: $R = s$
Why Hexagons?
Triangles are rigid. Squares fill space easily. Hexagons combine efficiency (max area per perimeter) with stacking ability, making them nature's favorite building block.