Dodecagon: Definition, Angles, Area, and Properties

TL;DR

A dodecagon is a polygon with 12 sides, 12 vertices, and 12 angles. Its interior angles always add to 1800°, a regular dodecagon has each interior angle equal to 150°, and it carries 54 diagonals. This article covers the definition, the four types, the angle, perimeter, and area formulas with their derivations, and the mistakes students make with the 12-sided shape.

What Is A Dodecagon?

A dodecagon is a closed two-dimensional polygon with 12 straight sides, 12 vertices, and 12 interior angles. The name comes from the Greek dodeka, meaning twelve, and gon, meaning sides or angles. Like every polygon, a dodecagon is defined by its side count alone — the sides can be equal or unequal, and the shape can bulge outward or cave inward.

When all 12 sides and all 12 angles are equal, it is a regular dodecagon, the version you meet most often. A dodecagon belongs to the same family as the pentagon and hexagon — just with more sides.

The Four Types Of Dodecagon

A dodecagon can be sorted by whether its sides and angles are equal, and by whether it bulges out or caves in.

• Regular dodecagon — all 12 sides equal and all 12 interior angles equal (150° each).
• Irregular dodecagon — sides and angles are not all equal; only the side count of 12 is fixed.
• Convex dodecagon — every interior angle is less than 180°, so no vertex points inward.
• Concave dodecagon — at least one interior angle is greater than 180° (a reflex angle), so one vertex caves inward.

Angles Of A Dodecagon

Sum of interior angles. Any polygon splits from one vertex into (n − 2) triangles, each worth 180°. The interior angles of a polygon therefore total:

[\text{Sum of interior angles} = (n - 2) \times 180°]

For a dodecagon, n = 12:

[(12 - 2) \times 180° = 10 \times 180° = 1800°]

A dodecagon splits into exactly 10 triangles, which is why the total is 10 lots of 180°.

Each interior angle of a regular dodecagon. Share the 1800° total across 12 equal corners:

[\text{Each interior angle} = \frac{1800°}{12} = 150°]

Each exterior angle of a regular dodecagon. Interior and exterior angles at a vertex make a straight line:

[\text{Each exterior angle} = 180° - 150° = 30°]

The exterior angles of any polygon add to 360°, which checks the answer: 360° ÷ 12 = 30°.

Variable glossary: n is the number of sides (12 here); (n − 2) is the number of triangles; 180° is one triangle's angle sum.

Perimeter Of A Dodecagon

The perimeter is the total distance around the shape — the sum of all 12 side lengths. For a regular dodecagon with side length s, every side is the same, so:

[P = 12s]

If a regular dodecagon has sides of 4 cm, its perimeter is 12 × 4 cm = 48 cm. For an irregular dodecagon you simply add the 12 individual side lengths; there is no shortcut.

Area Of A Dodecagon

For a regular dodecagon with side length s, the area formula is:

[A = 3(2 + \sqrt{3})s^{2}]

Where this comes from: a regular dodecagon can be cut into 12 identical isosceles triangles meeting at the center. Each triangle has area (\frac{1}{2} s^{2} \cot(15°)) and twelve of them yield the clean (3(2 + \sqrt{3}) s^{2}).

Variable glossary: A is the area, s is the length of one side, and 3(2 + √3) ≈ 11.196 is the constant that bundles the 12 triangles together.

How Many Diagonals Does A Dodecagon Have?

A diagonal joins two non-adjacent vertices. From each of the 12 vertices you can draw a diagonal to 9 others (you skip the vertex itself and its 2 neighbors). That counts every diagonal twice, so divide by 2:

[\text{Number of diagonals} = \frac{n(n-3)}{2} = \frac{12 \times 9}{2} = 54]

Examples Of The Dodecagon

Example 1

Find the sum of the interior angles of a dodecagon.

Sum = (n − 2) × 180°
Sum = (12 − 2) × 180°
Final answer: 1800°

Example 2

A student claims each interior angle of a regular dodecagon is 1800° ÷ 10 = 180°. Find the correct value. Take the wrong path first, because this is a common error.
Wrong attempt: 1800° ÷ 10 = 180°
Correct method: Final answer: 150°

Example 3

A regular dodecagon has a side length of 5 cm. Find its perimeter.
Final answer: 60 cm

Example 4

Find each exterior angle of a regular dodecagon.
Final answer: 30°

Example 5

Find the number of diagonals in a dodecagon.
Final answer: 54

Example 6

A regular dodecagon has a side of 4 cm. Find its area.
Final answer: ≈ 179.1 cm²

Why Twelve Sides Shows Up So Often

Twelve is a number that divides cleanly — by 2, 3, 4, and 6. A regular dodecagon's angles and layout enable practical applications in design and engineering.

Tripping Points To Avoid

Mistake 1: Dividing 1800° by 10 instead of 12.
Mistake 2: Applying 150° to an irregular dodecagon.
Mistake 3: Miscounting diagonals as n(n − 1) ÷ 2.

Conclusion

• A dodecagon has 12 sides, 12 vertices, and 12 angles.
• Its interior angles always add to 1800°.
• A regular dodecagon has each interior angle of 150° and each exterior angle of 30°.
• It carries 54 diagonals.
• The regular area is 3(2 + √3) s² and the perimeter is 12s.