Dodecagon Calculator - 12-Side Polygon Properties

A dodecagon calculator is a single-input tool that takes one side length of a regular 12-sided polygon and returns the area, perimeter, apothem, circumradius, interior and exterior angles, and the five distinct diagonal lengths, so you can stop pulling up six separate formulas for one shape. Enter a side in any common linear unit, pick the area unit you want, and the tool gives you all eleven numbers in one pass.

• • Geometry homework and exams: Verify a worked regular dodecagon problem, confirm the closed-form area, and check the diagonal lengths.
• • Crafts, tiles, and tabletop layouts: Estimate the area of a dodecagon-shaped patio, gazebo floor, tabletop, or inlay pattern from one side length.
• • Architecture and landscaping: Plan a dodecagon-shaped pool, fountain surround, or garden bed by turning the side length into the area the surface covers.
• • Engineering and product design: Get the apothem, circumradius, and longest diagonal (the diameter) to size the inscribed and circumscribed circles and the largest feature that fits inside a dodecagonal nut, bolt head, badge, or coin.

A regular dodecagon is a polygon with twelve equal sides and twelve equal interior angles, the natural next step after the square, hexagon, octagon, and decagon. Real dodecagons show up in clock faces, coin edge patterns, decorative tile layouts, and round-feeling buildings that still want a clean polygonal edge.

If you want the same set of regular-polygon properties for any n-sided shape other than a dodecagon, the polygon area calculator takes the same one-input approach and lets you change the number of sides.

The tool converts the side length to meters, evaluates the closed-form dodecagon area formula, derives the apothem, circumradius, perimeter, the two angle measures, and the five diagonal lengths, and then re-expresses the linear outputs in your chosen length unit and the area in your chosen area unit.

• s (side length): Length of one of the twelve equal sides, in the chosen length unit.
• R (circumradius): Distance from the center to a vertex, computed once and reused for every diagonal.
• k (diagonal index): Number of sides the diagonal spans; k = 2, 3, 4, 5, 6, with k = 6 giving the diameter 2R.

Each of the twelve triangles formed by the center, two adjacent vertices, and one side has the same height (the apothem) and the same base, so the area is 12 times one such triangle, giving A = (1/2) P r. The same R drives every diagonal since a chord spanning k sides subtends 2k * pi/n radians at the center.

According to Wolfram MathWorld, a regular dodecagon has 12 equal sides and area A = 3(2+sqrt(3)) s^2, which is approximately 11.1961524 s^2.

According to Math Open Reference, the area of a regular n-sided polygon is A = n s^2 / (4 tan(pi/n)), the apothem is r = s / (2 tan(pi/n)), and the circumradius is R = s / (2 sin(pi/n)).

The same regular-polygon formula with n = 10 instead of n = 12 drives the decagon area calculator, so the worked example, apothem ratio, and interior angle pattern are the closest comparison point for a ten-sided peer.

Four short ideas explain every number in the result panel and prevent common mix-ups when you work with a regular twelve-sided polygon.

These four ideas explain why the result panel returns more than just the area, and the symmetry of the 12-gon is what makes the diagonal ladder so short.

As the number of sides grows past 12, the regular polygon approaches a circle, and the circle calculator takes the same radius input and applies the limiting pi r^2 formula to the apothem or circumradius result.

Use the dodecagon calculator in five short steps, switching length or area units at any time without re-entering the side length.

1. 1 Enter the side length: Type the length of one side; the unit is set in the dropdown below.
2. 2 Pick the length unit: Select mm, cm, m, in, ft, or yd so the tool can convert the side length to meters internally. Perimeter, apothem, circumradius, and the five diagonals use the same unit.
3. 3 Pick the area unit: Select m^2, cm^2, ft^2, or in^2 for the area readout. The choice does not change the physical area, only the displayed number.
4. 4 Read the result panel: Watch the dodecagon area, perimeter, apothem, circumradius, interior and exterior angles, and the five diagonal lengths update in real time.
5. 5 Use the values in your project: Cross-check the apothem and circumradius against any drawing, use the longest diagonal to confirm the largest feature that fits, and switch units to compare against a plan measured in feet, inches, centimeters, or meters.

For a 1.2 m dodecagonal gazebo floor, type 1.2 and keep meters and square meters as the units. The result panel returns about 16.12 m^2 of inside area, 14.4 m of perimeter, 2.2392 m of apothem, 2.3182 m of circumradius, and a 4.6364 m longest diagonal.

To cross-check the interior and exterior angle formulas for a different regular polygon, the heptagon area calculator runs the same general n-sided area and angle logic on a seven-sided shape.

A dedicated dodecagon calculator removes the algebra from eleven tightly related numbers and keeps units consistent.

• • Eleven dodecagon values in one pass: Enter one side length and read off the area, perimeter, apothem, circumradius, interior and exterior angles, and all five diagonals at the same time.
• • Closed-form dodecagon formula: The tool evaluates the closed-form dodecagon area formula A = 3(2+sqrt(3)) s^2 directly, which results from the general regular-polygon formula A = n s^2 / (4 tan(pi/n)) at n = 12.
• • All five diagonals, not just one: The result panel reports the five distinct dodecagon diagonals across 2, 3, 4, 5, and 6 sides, so you can size chords, inlays, and through-features without a second tool.
• • Metric and imperial without re-entering: Switch the side length between mm, cm, m, in, ft, and yd, and the area output between m^2, cm^2, ft^2, and in^2, without re-typing the side length.
• • Apothem and circumradius included: The result panel reports both the inscribed and circumscribed circle radii, so you can size a coin, badge, or bolt head without a second tool.

For a quick sanity check on a 2D area from any shape, including the squares and rectangles that frame a dodecagon layout, the area calculator returns the area from the length and width in a single pass.

Three factors control the precision of every dodecagon result, and a few practical limits apply to any real twelve-sided shape.

• • The dodecagon calculator assumes a perfectly closed, planar, regular dodecagon. Real wood, tile, or metal shapes have rounded edges, joint gaps, or material thickness, so add a small waste margin above the calculated area.
• • Angles, apothem, circumradius, and diagonals assume a flat layout. If the dodecagon is curved onto a 3D surface, a different surface area tool is needed.
• • The calculator does not compute the dodecagon's side length backwards from a given area. Divide the area by 11.1961524 and take the square root, then enter the result into the side length field.

These factors and limits come straight from the standard regular dodecagon geometry described in elementary references. The tool is intended for planning, homework, and quick estimates, and it should be cross-checked against a project drawing or material list when the result drives a purchase order.

According to Wikipedia, a regular dodecagon has interior angle 150 degrees, exterior angle 30 degrees, apothem s(2+sqrt(3))/2, circumradius s(sqrt(6)+sqrt(2))/4, and five distinct diagonal lengths spanning 2, 3, 4, 5, and 6 sides.

When you only need the dodecagon's area and not its diagonals or angles, the dodecagon area calculator returns the same 3(2+sqrt(3)) s^2 result in a tighter, area-focused view.