Calculate the area, perimeter, apothem, circumradius, inradius, angles, and diagonals of a regular decagon from side length, circumradius, or apothem.
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.
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.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
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)
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.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
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.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
A regular decagon is a 10-sided polygon where all sides are equal in length and all interior angles are equal (144° each). Use this as a practical reminder before finalizing the result.
Area = (P × apothem) / 2, where P = 10s. Equivalently, A = (5s²/2)√(5 + 2√5). For s = 10, Area ≈ 769.42.
Each interior angle is (10 − 2) × 180° / 10 = 144°. The sum of all interior angles is 1,440°.
35 diagonals, computed as n(n−3)/2 = 10 × 7 / 2. Keep this note short and outcome-focused for reuse.
The apothem is the perpendicular distance from the center to the midpoint of any side: a = s / (2 tan(π/10)) ≈ 1.5388s.
In a regular decagon, the ratio of the circumradius to the side length equals the golden ratio φ ≈ 1.618. This connection arises because sin(π/10) = 1/(2φ).