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Pentagram Area Calculator

Geometry

golden ratio & shoelace exact

Quick Answer

Pentagram Area Calculator: golden ratio & shoelace exact. Pentagram Area Calculator uses standard geometry formulas to compute accurate results instantly.

Star size

cm
Star area112.26 cm²
Perimeter (10 arms)61.8 cm
Share of circumcircle35.7%

How it works

  1. 1Inner radius = R ÷ φ² = 10 ÷ 2.6180 ≈ 3.82 cm (φ = golden ratio (1+√5)/2)
  2. 2Area = 5 arm triangles + inner pentagon ≈ 1.1226 × R² ≈ 112.26 cm²
  3. 3Perimeter = 10 arms = 10R ÷ φ ≈ 61.8 cm (each arm = R/φ)
A regular pentagram has 36° tips, inner vertices on a circle of R/φ² ≈ 0.382R (φ ≈ 1.618); its area is always 35.7% of the circumcircle and perimeter 10R/φ ≈ 6.18R; one stroke segment (tip-dip-tip) = 2R/φ; results match the shoelace formula over all 10 vertices exactly; enter tip-to-tip distance for crafts and flags.

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How to Use

  1. Enter known values: fill sides, radius or angles in the Pentagram Area Calculator.
  2. Auto-solve the rest: all unknowns are solved instantly — no manual derivation.
  3. See formula and steps: the result panel shows the formula used and the substitution.
  4. Try another case: change any known value to solve again.

Key Points

  • Pentagram Area Calculator is a free online geometry tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: golden ratio & shoelace exact.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Pentagram Area Calculator formula

Pentagram Area Calculator uses standard geometry formulas to compute accurate results instantly.

FAQ

How do I use the Pentagram Area Calculator?
Open the page and fill in the input fields — results update automatically. golden ratio & shoelace exact.
Are the results accurate?
Yes. Standard formulas and conversion factors are used and verified.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. Everything runs locally in your browser.

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Source & Updates

Based on standard Euclidean geometry formulas. Last updated: 2025-07

Last updated:

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Maintained and periodically reviewed by the YiJianSuan editorial team — review process in ourEditorial Policy.

Cite this page: YiJianSuan. Pentagram Area Calculator[EB/OL]. https://www.yijiansuan.com/en/c/geometry/star-area

Results provided by this tool are for reference only and do not constitute professional advice.