Decagon Perimeter Calculator

A decagon is a polygon with exactly 10 sides and 10 vertices (corners). The word comes from the Greek "deka" meaning ten and "gonia" meaning angle. A regular decagon has all sides equal in length and all interior angles equal (144° each). An irregular decagon has sides of varying lengths and/or angles of varying measures. Decagons appear in architecture, design, and nature—the cross-section of a pencil is often a decagon, and some flowers and starfish exhibit decagonal symmetry.

For a regular decagon (all sides equal), the perimeter formula is simply P = 10 × s, where s is the side length. Example: If each side is 5 cm, then P = 10 × 5 = 50 cm. This works because a regular decagon has 10 identical sides. For an irregular decagon, you must add all 10 side lengths: P = s₁ + s₂ + s₃ + s₄ + s₅ + s₆ + s₇ + s₈ + s₉ + s₁₀.

In a regular decagon: Each interior angle = (n-2) × 180° / n = (10-2) × 180° / 10 = 8 × 180° / 10 = 144°. Each exterior angle = 360° / n = 360° / 10 = 36°. The sum of all interior angles = (n-2) × 180° = 8 × 180° = 1440°. The sum of all exterior angles always equals 360° for any convex polygon. Note: Interior angle + Exterior angle = 180° (they are supplementary), and 144° + 36° = 180°.

The area of a regular decagon can be calculated using: A = (5/2) × s² × cot(π/10), which simplifies to approximately A ≈ 7.6942 × s². Alternatively, using the apothem (a): A = (1/2) × P × a = (1/2) × 10s × a = 5sa. The apothem is the distance from the center to the midpoint of any side: a = s / (2 × tan(18°)) ≈ 1.5388 × s. Example: For s = 5 cm, A ≈ 7.6942 × 25 = 192.36 cm².

A decagon has 35 diagonals. The formula for the number of diagonals in any polygon is d = n(n-3)/2, where n is the number of sides. For a decagon: d = 10(10-3)/2 = 10 × 7 / 2 = 35. These diagonals connect non-adjacent vertices. In a regular decagon, diagonals come in several distinct lengths. The shortest diagonals connect vertices that are 2 positions apart, while the longest diagonal (diameter) passes through the center.

The apothem (inradius) is the distance from the center to the midpoint of any side—it's the radius of the inscribed circle. Formula: a = s / (2 × tan(π/10)) ≈ 1.5388 × s. The circumradius is the distance from the center to any vertex—it's the radius of the circumscribed circle. Formula: R = s / (2 × sin(π/10)) ≈ 1.6180 × s. Interestingly, the ratio R/a ≈ 1.0515 is related to the golden ratio. For s = 10 units: apothem ≈ 15.39 units, circumradius ≈ 16.18 units.

A regular decagon has: All 10 sides of equal length, all 10 interior angles equal (144° each), 10-fold rotational symmetry, 10 lines of symmetry. An irregular decagon has: Sides of potentially different lengths, angles of potentially different measures, reduced or no symmetry, same number of diagonals (35). Both types always have: 10 sides, 10 vertices, interior angles summing to 1440°, exterior angles summing to 360°. Most real-world decagons are irregular.

Decagons appear in many places: (1) Architecture - Some building floor plans, decorative windows, and tiles use decagonal shapes. (2) Currency - Some coins (like the Australian 50-cent coin) are dodecagonal (12-sided), though decagonal coins exist in some countries. (3) Games - Certain dice (d10) appear decagonal but are actually pentagonal trapezohedrons. (4) Nature - Some flowers exhibit 10-fold symmetry. (5) Engineering - Cross-sections of some pencils and structural components. (6) Design - Logos, patterns, and ornamental designs. (7) Astronomy - Some planetary orbit diagrams approximate decagons.