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.
Decagon Calculator
Calculate all properties of a regular decagon: perimeter, area, apothem, circumradius, interior angles, diagonals. Input by side, radius, apothem, area, or perimeter.
Decagon Calculator
units
Side Length⧉
10.0000
Each of the 10 equal sides = 10.0000 units
Perimeter⧉
100.0000
10 × 10.00 = 100.0000 units
Area⧉
769.4209
½ × perimeter × apothem = 769.4209 units²
Apothem (Inradius)⧉
15.3884
Distance from center to midpoint of a side = 15.3884 units
Circumradius⧉
16.1803
Distance from center to a vertex = 16.1803 units
Number of Diagonals⧉
35
n(n−3)/2 = 10×7/2 = 35 diagonals
Interior Angle⧉
144.00°
Each interior angle of a regular decagon = 144.00°
Sum of Interior Angles⧉
1,440°
(n−2)×180° = 8×180° = 1440°
Apothem vs Circumradius
Apothem
R
Apothem/Circumradius ratio = 0.9511 (≈ cos(18°) = 0.9511)
Regular Decagon
Green = apothem, Orange = circumradius, Red = center
Decagon Properties Reference
| Property | Formula | Value |
|---|---|---|
| Sides | n | 10 |
| Side length | s | 10.0000 units |
| Perimeter | 10s | 100.0000 units |
| Area | ½ × P × a | 769.4209 units² |
| Apothem | s / (2 tan(π/10)) | 15.3884 units |
| Circumradius | s / (2 sin(π/10)) | 16.1803 units |
| Interior angle | 144° | 144.00° |
| Exterior angle | 36° | 36.00° |
| Central angle | 36° | 36.00° |
| Diagonals | n(n−3)/2 | 35 |
| Long diagonal | 2R | 32.3607 units |
Planning notes, formulas, and examples
About the Decagon Calculator
A regular decagon is a ten-sided polygon with all sides equal and all interior angles equal to 144°. It is closely related to the regular pentagon — a pentagonal star (pentagram) is formed by connecting alternate vertices of a decagon. The regular decagon has special significance in geometry because its construction is linked to the golden ratio φ = (1 + √5)/2: the ratio of the circumradius to the side length equals 1/(2 sin 18°) ≈ 1.618, which is exactly φ. This calculator computes every metric of a regular decagon from any single measurement. Enter the side length, circumradius, apothem, area, or perimeter, and the tool derives all other properties: the 35 diagonals, interior and exterior angles, central angle, and the area formula A = (5/2)s²√(5 + 2√5). An interactive SVG shows the shape with the apothem and circumradius marked, and a ratio bar visualizes how the apothem compares to the circumradius. Decagons appear in architecture, tiling patterns, coin designs, and decorative art. The US Sacagawea dollar coin, for instance, has a decagonal border.
When This Page Helps
Decagon formulas involve trigonometric functions (sin π/10, tan π/10) and the golden ratio, making working by hand inconvenient. This calculator derives all properties from any single measurement — side, circumradius, apothem, area, or perimeter — so you never need to remember which formula to apply. It is useful for geometry homework, architecture and tiling pattern design, and anyone working with 10-sided shapes.
How to Use the Inputs
- Select what measurement you know: Side Length, Circumradius, Apothem, Area, or Perimeter.
- Enter the known value in the input field.
- Click a preset like "Side = 10" or "Side = 5" to load a common example.
- Review the perimeter, area, apothem, circumradius, interior angle, and diagonal count.
- Examine the decagon SVG diagram with apothem and circumradius marked.
- Optionally choose a measurement unit from the dropdown.
- Adjust Precision for the number of decimal places in all outputs.
Formula used
Perimeter = 10s
Area = (5/2)s² √(5 + 2√5) = ½ × P × apothem
Apothem = s / (2 tan(π/10))
Circumradius = s / (2 sin(π/10))
Interior angle = 144°Example Calculation
Result: Area ≈ 769.42, Perimeter = 100
For side = 10: Perimeter = 100, Area ≈ 769.42, Apothem ≈ 15.39, Circumradius ≈ 16.18, 35 diagonals.
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.
The Decagon and the Golden Ratio
The regular decagon is intimately connected to the golden ratio φ = (1 + √5)/2 ≈ 1.618. The circumradius-to-side ratio equals φ, and the diagonal-to-side ratio involves φ as well. A regular pentagon inscribed in the same circumcircle has side length equal to the decagon's long diagonal divided by φ. This relationship means decagon constructions appear naturally whenever the golden ratio is involved, such as in Penrose tilings and quasicrystals.
Decagons in Architecture and Design
Regular and semi-regular decagonal shapes appear in Islamic geometric art, where 10-fold symmetry creates intricate star patterns. The plan of Castel del Monte in Italy is built on an octagonal core, but many Islamic mosques and mausoleums use decagonal floor plans. Coin design frequently uses decagonal outlines — the Australian 50-cent coin is a dodecagon, while several commemorative coins use decagonal borders for their distinctive appearance and easy identification by touch.
Properties of Regular Polygons
The decagon is one member of the regular polygon family. As the number of sides n increases, the polygon approaches a circle: the ratio of inradius to circumradius approaches 1, and the area approaches πR². For the decagon (n=10), the interior angle is 144°, the sum of interior angles is 1440°, and the number of diagonals is 35. Comparing these values across different n helps students understand how polygon properties scale and converge to circle properties.
Sources & Methodology
Last updated:
Frequently Asked Questions
A polygon with 10 equal sides and 10 equal angles, each measuring 144 degrees.
A decagon has n(n−3)/2 = 10×7/2 = 35 diagonals.
The apothem is the perpendicular distance from the center to the midpoint of any side. It equals the inradius of the inscribed circle.
The ratio of the circumradius to the side length of a regular decagon equals the golden ratio φ ≈ 1.618.
Not by themselves — regular decagons cannot tile the plane alone. However, they appear in Penrose tilings and Islamic geometric patterns combined with other shapes.
The sum is (n−2)×180° = 8×180° = 1440° for any decagon, regular or irregular.
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