How do you find the area of a regular decagon?

For a regular decagon with side length s, the area is A = (5/2) × s² × √(5 + 2√5). This calculator applies that formula automatically.

Can I calculate a decagon from the circumscribed circle?

Yes. For a regular decagon, the circumradius R is R = s / (2 sin(π/10)). Enter R and the tool finds the side, area and perimeter.

What is special about a regular decagon?

A regular decagon is closely related to the golden ratio: the ratio between its circumradius and side is approximately 1.618, the golden ratio.

Geometry / Polygons

Decagon Calculator

Calculate the properties of a regular decagon (10-gon): area, perimeter, side, inradius (apothem), circumradius and golden-ratio relations. Enter one parameter — get all.

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Side length

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Perimeter

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Area

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Inradius (apothem)

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Circumradius

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Steps

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Regular decagon formulas

Let \( s \) be the side of a regular decagon (10 equal sides):

Perimeter: \( P = 10s \)

Area: \( A = \frac{5}{2} s^2 \sqrt{5 + 2\sqrt{5}} \approx 7.69420884\, s^2 \)

Inradius (apothem): \( r = \dfrac{s}{2 \tan(\pi/10)} \)

Circumradius: \( R = \dfrac{s}{2 \sin(\pi/10)} \approx 1.618\, s \)

The angle \( \pi/10 \) is 18°, so the decagon is closely linked to the golden ratio \( \varphi \approx 1.618 \).

From circumference to decagon

If you know the radius of the circle that the decagon is inscribed in, use \( s = 2R \sin(\pi/10) \) then apply the area formula.

FAQ

Does this work for irregular decagons?

No, because irregular decagons need more data (sides or coordinates). This tool follows the regular case like most online decagon calculators.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Formula (extracted text)

Perimeter: \( P = 10s \) Area: \( A = \frac{5}{2} s^2 \sqrt{5 + 2\sqrt{5}} \approx 7.69420884\, s^2 \) Inradius (apothem): \( r = \dfrac{s}{2 \tan(\pi/10)} \) Circumradius: \( R = \dfrac{s}{2 \sin(\pi/10)} \approx 1.618\, s \)

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
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