Decagon Formulas: Dimensions and Angles
A regular decagon with side length a follows predictable geometric relationships. Below are the essential formulas for computing its key properties:
Perimeter = 10 × a
Area = (10 × a² × cot(π ÷ 10)) ÷ 4 ≈ 7.694 × a²
Interior angle (α) = (10 − 2) × 180° ÷ 10 = 144°
Exterior angle (β) = 360° ÷ 10 = 36°
Circumcircle radius (R) = a ÷ (2 × sin(π ÷ 10)) ≈ 1.618 × a
Incircle radius / Apothem (r) = a ÷ (2 × tan(π ÷ 10)) ≈ 1.539 × a
a— Length of one sideα— Interior angle of the decagonβ— Exterior angle of the decagonR— Radius of the circumscribed circler— Radius of the inscribed circle (apothem)
Understanding Decagon Angles
Every interior angle in a regular decagon measures exactly 144°. Since all ten angles are equal, their sum totals 1,440°—a property that holds for any regular 10-sided polygon regardless of size.
The exterior angle, formed between one side and the extension of an adjacent side, measures 36°. The interior and exterior angles always sum to 180°, which is why 144° + 36° = 180°. These angle values depend entirely on the number of sides, never on the side length itself.
This means you could have a decagon with sides of 1 cm or 1 meter, and the angles remain unchanged at 144° and 36°.
The 35 Diagonals of a Regular Decagon
A decagon contains 35 diagonals in total—line segments connecting non-adjacent vertices. These diagonals vary in length depending on how many sides they span:
- Across 2 sides (10 diagonals): d₂ = (a ÷ 2) × √(10 + 2√5) ≈ 1.902 × a
- Across 3 sides (10 diagonals): d₃ = (a ÷ 2) × √(14 + 6√5) ≈ 2.618 × a
- Across 4 sides (10 diagonals): d₄ = a × √(5 + 2√5) ≈ 3.078 × a (equals the inscribed circle diameter)
- Across 5 sides (5 diagonals): d₅ = a × (1 + √5) ≈ 3.236 × a (equals the circumscribed circle diameter)
Notice that the longest diagonals span exactly half the decagon's perimeter, connecting vertices on opposite sides.
What Is the Apothem?
The apothem is the perpendicular distance from the decagon's center to the midpoint of any side. It represents the radius of the inscribed circle that fits perfectly inside the polygon, touching all ten sides at their midpoints.
For a regular decagon with side length a, the apothem r equals approximately 1.539 × a. You can use the apothem to calculate area using an alternative formula: Area = (Perimeter × Apothem) ÷ 2. This method works because you can divide the decagon into ten identical triangles, each with base a and height r.
Common Pitfalls When Working with Decagons
Avoid these frequent mistakes when calculating decagon properties:
- Confusing interior and exterior angles — Interior angles (144°) and exterior angles (36°) measure different things. The interior angle is what you see inside the polygon; the exterior angle is the supplement outside. They always sum to 180° for a regular polygon.
- Forgetting the apothem is a radius — The apothem is the perpendicular from center to side, identical to the incircle radius. It's not the same as the circumcircle radius. Using the wrong radius in your area calculation will give incorrect results.
- Assuming all diagonals are equal — Diagonals vary significantly in length (from ~1.9a to ~3.2a). Simply knowing 'the diagonal' isn't enough—you must specify which type. The longest diagonals (across 5 sides) are roughly 70% longer than the shortest ones (across 2 sides).
- Mixing up the formula variations — Area formulas like 2.5 × a² × √(5 + 2√5) are approximations. The exact formula uses cotangent. For precision engineering or design, use the precise formula rather than decimal approximations.