Dodecagon: Definition, Angles, Area, and Properties

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 and the one with the clean formulas below. A dodecagon belongs to the same family as the pentagon and hexagon — just with more sides — and all of them are part of the wider world of geometric shapes.

The Four Types Of Dodecagon

A dodecagon can be sorted two ways at once: 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

This is the heart of the topic, so build the numbers rather than memorising them.

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 $\tfrac{1}{2} s^2 \cot(15°)$, and twelve of them, after simplifying the cotangent of 15°, collapses to the clean $3(2 + \sqrt{3})s^2$. You do not need to redo the trigonometry each time — but knowing the area is "12 center triangles added up" is what makes the formula trustworthy rather than magic.

Variable glossary: A is the area, s is the length of one side, and $3(2 + \sqrt{3}) \approx 11.196$ is the constant that bundles the 12 triangles together.

How Many Diagonals Does A Dodecagon Have?

This is a favourite exam question. 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 neighbours). 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°

Sum = 10 × 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 the error that recurs.

Wrong attempt: the student divided the total 1800° by 10. The 10 is the number of triangles the shape splits into, which is where the 1800° total came from — it is not the number of angles.

1800° ÷ 10 = 180°

That answer is impossible: a 180° interior angle means a perfectly straight corner, which is no corner at all. The break is dividing by 10 instead of 12.

Correct method: a dodecagon has 12 equal corners, so divide by 12.

1800° ÷ 12 = 150°

Final answer: 150°

Example 3

A regular dodecagon has a side length of 5 cm. Find its perimeter.

All 12 sides are equal.

P = 12s

P = 12 × 5 cm

Final answer: 60 cm

Example 4

Find each exterior angle of a regular dodecagon.

The interior and exterior angle at a vertex add to 180°.

Each interior angle = 150°

Each exterior angle = 180° − 150°

Final answer: 30°

Example 5

Find the number of diagonals in a dodecagon.

Diagonals = n(n − 3) ÷ 2

Diagonals = 12 × (12 − 3) ÷ 2

Diagonals = 12 × 9 ÷ 2

Diagonals = 108 ÷ 2

Final answer: 54

Example 6

A regular dodecagon has a side of 4 cm. Find its area.

A = 3(2 + √3) s²

A = 3(2 + √3) × 4²

A = 3(2 + √3) × 16

A = 48(2 + √3)

A ≈ 48 × 3.732

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 — and that is exactly why the dodecagon keeps appearing.

A regular dodecagon's 150° corners and 30° exterior angles tile and rotate in ways a five- or seven-sided shape cannot. Twelve directions, each 30° apart, is also why a clock face, a compass rose, and many architectural floor plans settle on twelve-fold layouts. When engineers want a shape that is close to circular but still has flat, machinable edges — coins, bolt patterns, lens housings — twelve sides is the sweet spot: round enough to roll smoothly, flat enough to grip and orient. That blend of near-circle and hard edge is the dodecagon's whole reason for existing in the built world.

Tripping Points To Avoid

Mistake 1: Dividing 1800° by 10 instead of 12

Where it slips in: finding one interior angle of a regular dodecagon.

Don't do this: divide the 1800° total by 10. The 10 is the triangle count that produced the total; it is not the number of angles.

The correct way: divide by the number of angles, 12, giving 150°. The rusher who reaches for the 10 they just used gets 180° and should immediately notice a straight angle cannot be a polygon corner.

Mistake 2: Applying 150° to an irregular dodecagon

Where it slips in: any dodecagon that is not regular.

Don't do this: assume every interior angle is 150°. That value depends on all 12 angles being equal.

The correct way: only the 1800° sum is guaranteed for an irregular dodecagon. Add the known angles and subtract from 1800° to find a missing one. The memorizer who treats "dodecagon = 150°" as a definition rather than a regular-case result falls into this often.

Mistake 3: Miscounting diagonals as n(n − 1) ÷ 2

Where it slips in: the diagonals formula.

Don't do this: use n(n − 1) ÷ 2, which is the count of all connecting segments including the sides.

The correct way: use n(n − 3) ÷ 2 — the "− 3" removes the vertex itself and its two adjacent sides, leaving only true diagonals. For n = 12 that is 54, not the 66 the wrong formula gives.

Conclusion

• A dodecagon has 12 sides, 12 vertices, and 12 angles.

• Its interior angles always add to 1800°, from (n − 2) × 180° with n = 12.

• A regular dodecagon has each interior angle = 150° and each exterior angle = 30°.

• It carries 54 diagonals, from n(n − 3) ÷ 2.

• The regular area is 3(2 + √3) s² and the perimeter is 12s.

A Practical Next Step

Work through the exercises above in order: confirm the 1800° total by splitting a dodecagon into 10 triangles, then derive the 150° and 30° angles yourself before reaching for the table. If the diagonals count trips you, return to "How many diagonals" and re-derive the n(n − 3) ÷ 2 logic.

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