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.
Property | Formula | Regular value (n = 12) |
|---|---|---|
Sum of interior angles | (n − 2) × 180° | 1800° |
Each interior angle | 1800° ÷ 12 | 150° |
Each exterior angle | 180° − 150° | 30° |
Number of diagonals | n(n − 3) ÷ 2 | 54 |
Perimeter | 12s | 12s |
Area | 3(2 + √3) s² | ≈ 11.196 s² |
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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