Decagon Area & Properties Calculator

Calculate the area, perimeter, apothem, circumradius, inradius, angles, and diagonals of a regular decagon from side length, circumradius, or apothem.

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Planning notes, formulas, and examples

About the Decagon Area & Properties Calculator

A regular decagon is a polygon with 10 equal sides and 10 equal angles. Each interior angle measures exactly 144°, and each exterior angle is 36°. The decagon occupies a special place in geometry because of its connection to the golden ratio (φ ≈ 1.618): the ratio of the circumradius to the side length of a regular decagon is exactly φ.

The area of a regular decagon with side length s is (5s²/2) × √(5 + 2√5), which can also be expressed as (P × apothem) / 2, where P is the perimeter and the apothem is the perpendicular distance from the center to the midpoint of any side. The apothem equals s / (2 tan(π/10)).

A regular decagon has 35 diagonals — computed as n(n−3)/2 = 10×7/2. Its symmetry group has order 20 (10 rotations and 10 reflections), making it one of the most symmetrical polygons. Decagonal shapes appear in architecture (some domed ceilings), coin design (Australian 50-cent coin), and tiling patterns.

This calculator computes every property of a regular decagon from any of three inputs: side length, circumradius, or apothem. It displays area, perimeter, all angles, the number of diagonals, and provides visual comparisons and reference tables including a comparison with other regular polygons.

When This Page Helps

The Decagon Area & Properties Calculator is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Area, Perimeter, Side Length in one pass, with conversions and derived values shown together.

How to Use the Inputs

  1. Choose what you know: side length, circumradius, or apothem.
  2. Select a measurement unit.
  3. Enter the known value.
  4. Or click a preset to load common decagon sizes.
  5. View area, perimeter, apothem, circumradius, and all angles.
  6. Compare the side, apothem, and circumradius visually.
  7. Review the polygon comparison table for context with other shapes.
Formula used
Area: A = (P × apothem) / 2, or A = (5s²/2)√(5 + 2√5) Perimeter: P = 10s Apothem: a = s / (2 tan(π/10)) Circumradius: R = s / (2 sin(π/10)) Inradius: r = apothem Interior angle: (n−2)×180°/n = 144° Exterior angle: 360°/n = 36° Diagonals: n(n−3)/2 = 35 R/s = φ ≈ 1.618 (golden ratio)

Example Calculation

Result: Area ≈ 769.42, Perimeter = 100, Apothem ≈ 15.39, Circumradius ≈ 16.18

For s = 10: Perimeter = 100. Apothem = 10/(2 tan(18°)) ≈ 15.39. Area = (100 × 15.39)/2 ≈ 769.42. Circumradius = 10/(2 sin(18°)) ≈ 16.18, which equals 10φ.

Tips & Best Practices

  • The ratio R/s for a regular decagon equals the golden ratio φ ≈ 1.618 — a unique property among regular polygons.
  • A decagon has 35 diagonals. That's more than a hexagon (9) or octagon (20) but fewer than a dodecagon (54).
  • As the number of sides increases, a regular polygon's area approaches that of a circle. A decagon already captures ~98% of its circumscribed circle area.
  • The interior angle sum of a decagon is 1,440° = (10 − 2) × 180°.

How This Decagon Area & Properties Calculator Works

Where It Helps In Practice

Decagon Area & Properties Calculator calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.

Accuracy And Setup Tips

Sources & Methodology

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Frequently Asked Questions

  • A regular decagon is a 10-sided polygon where all sides are equal in length and all interior angles are equal (144° each).

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