Free Decagon Calculator

Regular Decagon: Key Properties and Uses

A regular decagon is a polygon with ten equal sides and ten equal angles. The Regular Decagon Calculator simplifies the computation of its geometric properties—side length, perimeter, area, circumradius, inradius (apothem), diagonal lengths, and interior and exterior angles—based on any one known value. This guide covers the mathematical formulas behind these measurements and explains how to use the calculator effectively.

Perimeter, Area, and Radii

When the side length $a$ is known, the following formulas apply:

• Perimeter: $P = 10a$
• Area: $$A = \frac{5}{2} a^{2} \sqrt{5 + 2\sqrt{5}} \approx 7.694 a^{2}$$
• Circumradius (radius of the circumscribed circle): $$R = \frac{a}{2}(1 + \sqrt{5}) \approx 1.618 a$$
• Inradius (radius of the inscribed circle, also called the apothem): $$r = \frac{a}{2} \sqrt{5 + 2\sqrt{5}} \approx 1.539 a$$

The apothem is the perpendicular distance from the center to any side.

Interior and Exterior Angles

• Each interior angle measures $144^{\circ}$.
• The sum of all ten interior angles is $1440^{\circ}$.
• Each exterior angle is $36^{\circ}$, because $144^{\circ} + 36^{\circ} = 180^{\circ}$.

These angle values are fixed for a regular decagon and do not depend on the side length.

Diagonals of a Decagon

A regular decagon has 35 diagonals in total. Depending on how many sides a diagonal skips, its length varies:

Sides skipped	Number of diagonals	Length formula	Approximate value
2	10	$\dfrac{a}{2}\sqrt{10+2\sqrt{5}}$	$1.902a$
3	10	$\dfrac{a}{2}\sqrt{14+6\sqrt{5}}$	$2.618a$
4	10	$a\sqrt{5+2\sqrt{5}}$	$3.078a$
5	5	$a(1+\sqrt{5})$	$3.236a$

These diagonals appear in patterns within the decagon and are useful in design and analysis.

How to Use the Decagon Calculator

The Decagon Calculator accepts any single known value: side length, perimeter, area, circumradius, inradius, or one of the diagonal lengths. Once entered, it instantly computes all other properties. The tool also supports unit adjustments, and you can check the “Try other regular polygons” option to explore polygons with different numbers of sides. This makes it a versatile Decagon Side Length Calculator, Area Calculator, and Perimeter Calculator all in one.

Summary

Whether you are solving a geometry problem, designing a decagonal element, or simply curious about 10-sided shapes, the Regular Decagon Calculator provides fast, accurate results. The formulas presented here give you the foundation needed to understand its calculations and interpret the outputs.