Pentagram calculator and formula

This function calculates various parameters of a pentagram.

To perform the calculation, select the property you know from the pull down menu and enter its value. Then click the 'Calculate' button.

Pentagram calculator

Input

Decimal places

Results

Pentagon edge a

Chord length l

Long chord slice b

Short chord slice c

Perimeter P

Area A

Circle radius r

Formulas for the pentagram

Golden cut

\(\displaystyle Φ=\frac{1+\sqrt{5}}{2} \;≈\;1.618\)

Edge length of the pentagon

\(\displaystyle a = b + c\) \(\displaystyle \;\;\;\; = \frac{l}{Φ}\) \(\displaystyle \;\;\;\; ≈2.618 ·l\)

Chord length l

\(\displaystyle l=Φ · a \;\;\;\;≈\;1.618 · a\)

Long chord slice b

\(\displaystyle b=\frac{a}{Φ} \)

Short chord slice (inner pentagon) c

\(\displaystyle c=\frac{a}{Φ^2} \;\;\;\; = \frac{b}{Φ}\)

Area

\(\displaystyle A=\frac{1}{2} · \sqrt{25-10· \sqrt{5}}·a^2 \;\;\;\;≈0.812 ·a^2\)

Perimeter

\(\displaystyle P= 10·b\)

Circumcircle radius

\(\displaystyle A=\frac{1}{10} · \sqrt{50+10· \sqrt{5}}·a \;\;\;\;≈0.851 ·a\)

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