Decagon Calculator

Properties of a Regular Decagon (10-Sided Polygon)

A decagon is a ten-sided polygon ($n = 10$). In a regular decagon, all 10 sides have equal length $a$, all 10 interior angles equal $144^\circ$, and all 10 vertices lie on a common circumscribed circle. The geometry of a regular decagon is intimately connected with the Golden Ratio ($\phi = \frac{1+\sqrt{5}}{2} \approx 1.6180339887$).

Decagon Formulas Reference

1. Interior, Exterior, and Central Angles

• Interior Angle ($\alpha$): $\alpha = \frac{(10 - 2) \times 180^\circ}{10} = 144^\circ = \frac{4\pi}{5}\text{ rad}$
• Central Angle ($\theta$): $\theta = \frac{360^\circ}{10} = 36^\circ = \frac{\pi}{5}\text{ rad}$
• Exterior Angle ($\beta$): $\beta = 180^\circ - 144^\circ = 36^\circ = \frac{\pi}{5}\text{ rad}$
• Sum of Interior Angles ($S$): $S = (10 - 2) \times 180^\circ = 1440^\circ = 8\pi\text{ rad}$
• Total Number of Diagonals ($D$): $D = \frac{n(n - 3)}{2} = \frac{10 \times 7}{2} = 35\text{ diagonals}$

2. Area of a Regular Decagon

The total surface area $A$ of a regular decagon with side length $a$ is calculated as:

$$A = \frac{5}{2} a^2 \cot\left(\frac{\pi}{10}\right) = \frac{5}{2} a^2 \sqrt{5 + 2\sqrt{5}} \approx 7.69420884 \times a^2$$

Alternatively, using perimeter $P = 10a$ and inradius (apothem) $r$:

$$A = \frac{1}{2} P \cdot r = 5 a r$$

3. Inradius (Apothem) and Circumradius

The inradius ($r$) is the perpendicular distance from the center to the midpoint of any side:

$$r = \frac{a}{2 \tan(18^\circ)} = \frac{a}{2} \sqrt{5 + 2\sqrt{5}} \approx 1.53884177 \times a$$

The circumradius ($R$) is the radius of the circle passing through all 10 vertices:

$$R = \frac{a}{2 \sin(18^\circ)} = a \times \phi = a \left(\frac{1 + \sqrt{5}}{2}\right) \approx 1.61803399 \times a$$

Diagonals of a Regular Decagon

A regular decagon contains 35 diagonals of four distinct lengths: