Arc Length Calculator
Calculate arc length, sector area, chord length, and arc-to-chord ratio from radius and central angle. Includes preset common angles, a properties table, and an arc diagram.
Dodecagon Calculator
Calculate regular dodecagon (12-sided polygon) properties: perimeter, area, diagonals, interior/exterior angles, apothem, inradius, circumradius. Presets, properties table, and polygon visual.
Dodecagon Calculator
Side Length⧉
5.0000 units
Length of one of the 12 equal sides
Perimeter⧉
60.0000 units
12 × 5.00 = 60.00
Area⧉
279.9038 units²
3(2+√3)s² ≈ 11.196 × 5.00²
Apothem (Inradius)⧉
9.3301 units
Distance from center to midpoint of a side
Circumradius⧉
9.6593 units
Distance from center to a vertex
Interior Angle⧉
150.00°
(12−2)×180°/12 = 150°
Exterior Angle⧉
30.00°
360°/12 = 30°
Number of Diagonals⧉
54
12×(12−3)/2 = 54
Dodecagon
Red dashed = circumradius, Green dashed = apothem
Diagonal Lengths
d2 (span 2)9.6593 units
d3 (span 3)13.6603 units
d4 (span 4)16.7303 units
d5 (span 5)18.6603 units
d6 (span 6)19.3185 units
Polygon Comparison (same side length)
| Polygon | Sides | Int. Angle | Perimeter | Area | Diagonals |
|---|---|---|---|---|---|
| 3-gon | 3 | 60.0° | 15.00 | 10.83 | 0 |
| 4-gon | 4 | 90.0° | 20.00 | 25.00 | 2 |
| 5-gon | 5 | 108.0° | 25.00 | 43.01 | 5 |
| 6-gon | 6 | 120.0° | 30.00 | 64.95 | 9 |
| 8-gon | 8 | 135.0° | 40.00 | 120.71 | 20 |
| 10-gon | 10 | 144.0° | 50.00 | 192.36 | 35 |
| 12-gon | 12 | 150.0° | 60.00 | 279.90 | 54 |
| 15-gon | 15 | 156.0° | 75.00 | 441.06 | 90 |
| 20-gon | 20 | 162.0° | 100.00 | 789.22 | 170 |
Planning notes, formulas, and examples
About the Dodecagon Calculator
A regular dodecagon is a twelve-sided polygon with all sides equal and all interior angles equal. It appears in everyday life as the shape of many clock faces, some coins (UK pound coin), and stop-sign-style designs. Each interior angle of a regular dodecagon measures 150°, and the sum of all interior angles is 1800°. The area of a regular dodecagon with side length s is 3(2 + √3)s², which is approximately 11.196s². The number of diagonals is 54. This calculator takes the side length as primary input (or you can specify the circumradius, inradius/apothem, or area and it will derive the side length). It outputs the perimeter, area, apothem (inradius), circumradius, diagonal lengths, interior and exterior angles, and more. A polygon visual draws the dodecagon with labeled vertices, and a properties comparison table shows how the dodecagon compares to other regular polygons from triangle to icosagon.
When This Page Helps
Dodecagon formulas involve trigonometric expressions with π/12 (sin 15° and tan 15°), making working by hand tedious. This calculator derives all properties from any single measurement and includes a polygon comparison table so you can see how the dodecagon relates to other regular polygons. It is useful for geometry students, tile pattern designers, and anyone working with 12-fold symmetry.
How to Use the Inputs
- Select the input mode: Side Length, Circumradius, Apothem, or Area.
- Enter the known value in the input field.
- Click a preset like "s = 5" or "s = 10" to load a common example.
- Review the perimeter, area, apothem, circumradius, interior/exterior angles, and diagonal count.
- Examine the dodecagon SVG diagram with labeled vertices.
- Compare properties with other polygons (triangle through 20-gon) in the comparison table.
- Adjust Precision to control the number of decimal places displayed.
Formula used
Interior angle = (n−2)×180°/n = 150° (n=12)
Area = 3(2+√3)s² ≈ 11.196s²
Perimeter = 12s
Apothem = s/(2 tan(π/12))
Circumradius = s/(2 sin(π/12))
Diagonals = n(n−3)/2 = 54Example Calculation
Result: Area ≈ 279.9, Perimeter = 60
A regular dodecagon with side 5: Area = 3(2+√3)×25 ≈ 279.9, Perimeter = 60, Apothem ≈ 9.33, Circumradius ≈ 9.66.
Tips & Best Practices
- Check that all inputs use the same scale and assumptions before trusting the result.
- Compare the answer with the worked example or a rough estimate to catch entry mistakes.
Dodecagons in the Real World
The regular dodecagon appears more often than you might expect. The modern UK one-pound coin has a 12-sided shape for security and tactile recognition. Many clock faces approximate a dodecagonal layout with 12 hour markers. In architecture, 12-fold rosettes are common in Gothic cathedral rose windows. The dodecagon also tiles the plane in combination with triangles and squares (the "3-4-6-12" Archimedean tiling).
The Geometry of 12-Fold Symmetry
A regular dodecagon has 12 lines of symmetry and rotational symmetry of order 12 (30° increments). Its interior angle of 150° means three dodecagons cannot meet at a vertex (3×150° = 450° > 360°), so regular dodecagons alone cannot tile the plane. Its area formula 3(2+√3)s² ≈ 11.196s² makes it very close to a circle (for comparison, a circle of the same circumradius has area πR²). As n increases, the n-gon area approaches πR², and the dodecagon is already 98.86% of the circular area.
Diagonal Structure
A dodecagon has 54 diagonals with 5 distinct diagonal lengths (spanning 2, 3, 4, 5, and 6 vertices). The longest diagonal (spanning 6 vertices) equals the diameter (2R). These diagonals intersect to create intricate star patterns — the most famous being the Star of David pattern formed by two overlapping hexagons inscribed in the dodecagon. This rich diagonal structure is exploited in Islamic geometric art and quilt patterns.
Sources & Methodology
Last updated:
Frequently Asked Questions
A dodecagon has 12 sides. The prefix "dodeca-" comes from the Greek word for twelve.
Each interior angle is 150°. The sum of all interior angles is 1800°.
A dodecagon has 54 diagonals, calculated as n(n−3)/2 = 12×9/2 = 54.
The apothem is the distance from the center to the midpoint of a side. For a regular dodecagon, apothem = s/(2 tan(π/12)).
Area = 3(2+√3)s², or equivalently (1/2) × perimeter × apothem = (1/2) × 12s × apothem.
Clock faces, some coins (old UK three-pence and the modern £1 coin), architectural elements, and tile patterns.
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