Hexagon Calculator

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: Side Length 10

Problem:s = 10

Solution:Area = (3√3/2)×100 ≈ 2.598×100

Result:259.81

Example 2: Perimeter

Problem:s = 5

Solution:Perimeter = 6 × 5

Result:30

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.

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 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).

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.

History

The hexagon has been studied since antiquity due to its prevalence in nature and efficiency. The "Honeycomb Conjecture"—that the hexagon tiles a plane with minimal perimeter—was proposed by Marcus Terentius Varro in 36 BC but was only mathematically proven by Thomas Hales in 1999. Euclid and Archimedes studied the properties of regular polygons. The hexagon holds a special place because it can be constructed using only a compass and straightedge, as its side length is equal to the radius of its circumscribed circle.