Calculate all properties of a regular decagon: perimeter, area, apothem, circumradius, interior angles, diagonals. Input by side, radius, apothem, area, or perimeter.
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.
Decagon formulas involve trigonometric functions (sin π/10, tan π/10) and the golden ratio, making manual computation 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.
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°
Result: Area ≈ 769.42, Perimeter = 100
For side = 10: Perimeter = 100, Area ≈ 769.42, Apothem ≈ 15.39, Circumradius ≈ 16.18, 35 diagonals.
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.
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.
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.
A polygon with 10 equal sides and 10 equal angles, each measuring 144 degrees. Use this as a practical reminder before finalizing the result.
A decagon has n(n−3)/2 = 10×7/2 = 35 diagonals. Keep this note short and outcome-focused for reuse.
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. Apply this check where your workflow is most sensitive.