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Hexagon

Hexagon

A polygon with six sides.

 

A hexagon with six sides of slightly unequal lengths, forming an irregular closed polygon shape.

Hexagon

A regular hexagon with six equal sides and angles, representing a regular polygon shape.

Regular Hexagon

Key Formula

S=(n−2)×180°where n=6⇒S=720°S = (n - 2) \times 180°\quad\text{where } n = 6\quad\Rightarrow\quad S = 720°
Where:
  • SS = Sum of the interior angles of the hexagon
  • nn = Number of sides (6 for a hexagon)

Worked Example

Problem: A regular hexagon has a side length of 10 cm. Find the perimeter, each interior angle, and the area.
Step 1: Find the perimeter. A regular hexagon has 6 equal sides, so multiply the side length by 6.
P=6×10=60 cmP = 6 \times 10 = 60 \text{ cm}
Step 2: Find each interior angle. The sum of interior angles is 720°. In a regular hexagon every angle is the same, so divide by 6.
Each angle=720°6=120°\text{Each angle} = \frac{720°}{6} = 120°
Step 3: Find the area using the regular-hexagon area formula. A regular hexagon can be split into 6 equilateral triangles.
A=332 s2=332×102=332×100=1503A = \frac{3\sqrt{3}}{2}\,s^2 = \frac{3\sqrt{3}}{2}\times 10^2 = \frac{3\sqrt{3}}{2}\times 100 = 150\sqrt{3}
Step 4: Approximate the area to a decimal value.
A≈150×1.732=259.8 cm2A \approx 150 \times 1.732 = 259.8 \text{ cm}^2
Answer: The perimeter is 60 cm, each interior angle is 120°, and the area is approximately 259.8 cm².

Another Example

Problem: Three angles of an irregular hexagon measure 100°, 130°, and 140°. The remaining three angles are all equal. Find the measure of each remaining angle.
Step 1: Use the interior angle sum formula for a hexagon.
S=(6−2)×180°=720°S = (6 - 2) \times 180° = 720°
Step 2: Add the three known angles.
100°+130°+140°=370°100° + 130° + 140° = 370°
Step 3: Subtract from 720° to find the total of the three unknown angles.
720°−370°=350°720° - 370° = 350°
Step 4: Since the three remaining angles are equal, divide by 3.
350°3≈116.67°\frac{350°}{3} \approx 116.67°
Answer: Each of the three remaining angles measures approximately 116.67° (or exactly 350°/3).

Frequently Asked Questions

How many diagonals does a hexagon have?
A hexagon has 9 diagonals. You can calculate this with the diagonal formula: D = n(n − 3)/2 = 6(6 − 3)/2 = 9.
Why are hexagons so common in nature?
Regular hexagons tile a flat surface with no gaps while enclosing the most area for a given perimeter among tilings. This is why honeybees build hexagonal cells — it is the most material-efficient way to divide a surface into equal areas. Basalt columns and bubble rafts also naturally form hexagonal patterns for similar efficiency reasons.

Hexagon (6 sides) vs. Pentagon (5 sides)

A hexagon has 6 sides with an interior angle sum of 720°, while a pentagon has 5 sides with an interior angle sum of 540°. In their regular forms, each interior angle of a hexagon is 120° and each interior angle of a pentagon is 108°. Regular hexagons can perfectly tessellate a plane; regular pentagons cannot.

Why It Matters

Hexagons appear throughout science and engineering. Honeycomb structures use hexagonal cells for maximum strength with minimum material, a principle applied in aerospace panels and packaging. In chemistry, the benzene ring — a hexagonal arrangement of carbon atoms — is foundational to organic chemistry. Hex grids are also widely used in game design and geographic mapping.

Common Mistakes

Mistake: Using 180° × 6 = 1080° for the interior angle sum instead of the correct formula (n − 2) × 180°.
Correction: Always subtract 2 from the number of sides first: (6 − 2) × 180° = 720°. The formula accounts for the fact that a polygon can be divided into (n − 2) triangles.
Mistake: Assuming all hexagons are regular (equal sides and equal angles).
Correction: A hexagon only needs six sides. The sides can have different lengths and the angles can differ, as long as the angles still sum to 720°. Only a regular hexagon has all sides and all angles equal.

Related Terms

  • Polygon — General term for any closed multi-sided figure
  • Side of a Polygon — Each straight segment forming the hexagon
  • Regular Polygon — A polygon with all sides and angles equal
  • Pentagon — The five-sided polygon, one fewer side
  • Octagon — The eight-sided polygon, two more sides
  • Interior Angle — Each angle inside the hexagon at a vertex
  • Tessellation — Regular hexagons tile a plane with no gaps