Dodecagon Calculator
A regular dodecagon is a twelve-sided polygon with all sides and all interior angles equal. Enter any one known value and get every other property instantly, including all five distinct diagonal lengths, incircle and circumcircle areas, and the total diagonal count.
Formula
Worked example
For a = 5 cm: P = 60 cm, A = 3(2+√3)×25 ≈ 279.90 cm², r ≈ 9.33 cm, R ≈ 9.66 cm. The five diagonals are d2 ≈ 9.66 cm, d3 ≈ 13.66 cm, d4 ≈ 16.73 cm, d5 ≈ 18.66 cm, d6 ≈ 19.32 cm. Incircle area ≈ 273.5 cm², circumcircle area ≈ 293.1 cm². The dodecagon fills 95.5% of its circumcircle. Reverse-solve: area 279.90 → a = √(279.90/11.196) = 5 cm.
How the dodecagon area formula works
A regular dodecagon can be divided from its centre into twelve identical isosceles triangles, one for each side. Each triangle has a base equal to the side length a and a height equal to the apothem r, the perpendicular distance from the centre to the midpoint of a side. Adding the twelve triangle areas and simplifying the trigonometry gives the compact closed form A = 3(2+√3)a², where the constant 3(2+√3) is approximately 11.196. Because the side length is squared, area responds far more strongly to a change in side than the perimeter does: doubling the side multiplies the area by four while only doubling the perimeter. The perimeter itself is the simplest property of all - twelve equal edges give P = 12a.
Apothem, circumradius, and the two circles
Two radii fully describe the size of a regular polygon. The apothem r (also called the inradius) reaches from the centre to the midpoint of a side and equals (a/2)(2+√3) ≈ 1.866a; it is the radius of the largest inscribed circle. The circumradius R reaches from the centre to a vertex and equals a/(2·sin 15°) ≈ 1.932a; it defines the circumscribed circle passing through every corner. The incircle has area pi·r² and the circumcircle has area pi·R². A regular dodecagon fills about 95.5% of its circumcircle, which is why it looks so close to a circle - only a small amount of the surrounding circle is wasted.
The five diagonal lengths and diagonal count
A regular dodecagon has 54 diagonals in total, calculated with the formula n(n-3)/2 = 12×9/2 = 54. Because of the polygon's symmetry, these 54 diagonals fall into exactly five distinct lengths. The shortest diagonal d2 connects vertices two steps apart and equals approximately 1.932a - the same length as the circumradius. The next diagonal d3 skips two vertices and measures about 2.732a. Moving further, d4 ≈ 3.346a, d5 ≈ 3.732a, and the longest diagonal d6 passes directly through the centre and equals 2R ≈ 3.864a. Each diagonal length follows the general chord formula d_k = a × sin(k×15°) / sin(15°), where k is the number of steps between connected vertices.
Reverse-solve: start from any known measurement
This calculator lets you start from any single known value. Enter the perimeter and the side is found by dividing by 12. Enter the area and the side is recovered from a = √(A / 3(2+√3)). Enter the circumradius R and the side follows as a = 2R·sin(15°). Enter the apothem r and the side is a = 2r / (2+√3). All five input modes produce the same full set of outputs, so whichever dimension you can measure directly - whether it is a floor radius or the distance across flats - the calculator fills in everything else.
Where dodecagons appear
The twelve-sided shape appears wherever a near-circular outline is needed from straight edges. The British twelve-sided one-pound coin uses a dodecagonal profile so the coin is identifiable by touch yet still moves through machines. Architects use dodecagonal floor plans for gazebos, bandstands, and observation towers. Tile designers exploit the fact that dodecagons combine neatly with squares and equilateral triangles in semiregular Archimedean tessellations. In engineering, twelve-sided bolt heads and nuts appear in specialised fasteners. In each case the formulas here let you move from a single known dimension to the full geometry in one step.
Regular dodecagon: all properties at a glance
| Property | Symbol | Formula (side = a) | Value when a = 1 |
|---|---|---|---|
| Perimeter | P | 12a | 12.0000 |
| Area | A | 3(2+√3)·a² | 11.1962 |
| Apothem (inradius) | r | (a/2)(2+√3) | 1.86603 |
| Circumradius | R | a / (2·sin 15°) | 1.93185 |
| Shortest diagonal | d2 | a·sin(30°)/sin(15°) | 1.93185 |
| Diagonal | d3 | a·sin(45°)/sin(15°) | 2.73205 |
| Diagonal | d4 | a·sin(60°)/sin(15°) | 3.34607 |
| Diagonal | d5 | a·sin(75°)/sin(15°) | 3.73205 |
| Longest diagonal | d6 | a/sin(15°) = 2R | 3.86370 |
| Interior angle | alpha | (n-2)·180°/n | 150° |
| Exterior angle | beta | 360°/n | 30° |
| Number of diagonals | - | n(n-3)/2 | 54 |
| Incircle area | Ai | pi·r² | 10.9392 |
| Circumcircle area | Ac | pi·R² | 11.7246 |
All constants are exact or rounded to 6 significant figures. The calculator keeps full floating-point precision.
Frequently asked questions
What is a regular dodecagon?
A regular dodecagon is a polygon with twelve sides of equal length and twelve equal interior angles of 150° each. Its high symmetry means a single side length, radius, or area determines every other measurement - area, perimeter, apothem, circumradius, and all five diagonal lengths all follow from that one value.
What is the area of a regular dodecagon with side a?
The area of a regular dodecagon is A = 3(2+√3)·a², which is approximately 11.196·a². For example, a side of 10 gives an area of about 1119.6 square units. The formula comes from splitting the shape into twelve identical triangles meeting at the centre.
How many diagonals does a dodecagon have, and how many lengths?
A regular dodecagon has 54 diagonals in total, from the formula n(n-3)/2 = 12×9/2 = 54. Due to symmetry, these 54 diagonals fall into only five distinct lengths. The shortest connects vertices two steps apart (approximately 1.932 times the side), and the longest passes through the centre connecting opposite vertices (approximately 3.864 times the side).
How do I find the side length if I only know the area?
Rearrange A = 3(2+√3)·a² to get a = √(A / 3(2+√3)) = √(A / 11.196). Use the "Solve from: Area" mode in this calculator and enter the area directly. For example, an area of 279.90 gives a side of exactly 5.
How many degrees are in a dodecagon?
The interior angles of any dodecagon sum to (12-2)×180° = 1800°. In a regular dodecagon those degrees are shared equally, so each of the twelve interior angles measures 150°, and each exterior angle measures 30°.
What is the difference between the apothem and the circumradius?
The apothem (inradius r) is the distance from the centre to the midpoint of a side - it is the radius of the inscribed circle that fits inside the dodecagon touching every side. The circumradius R is the distance from the centre to a vertex - it is the radius of the circumscribed circle passing through all twelve corners. For a regular dodecagon, R ≈ 1.932a and r ≈ 1.866a, so R is slightly larger than r.