Pentagram calculator
Calculator and formulas for calculating a pentagram
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.
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Formulas for the pentagram
Golden ratio (\(\small{φ}\))
\(\displaystyle ?=\frac{1+\sqrt{5}}{2} \;≈\;1.618\)
Within the pentagram you can find the four lengths \(l, a, b, \text{and} \ c\). The ratio of one of these lengths to the next smaller one results in the golden ratio.
Side length of the outer pentagon (\(\small{a}\))
\(\displaystyle a = b + c\) \(\displaystyle \;\;\;\; = \frac{l}{Φ}\) \(\displaystyle \;\;\;\; ≈2.618 ·l\)
Overall length (\(\small{l}\))
\(\displaystyle l=Φ · a \;\;\;\;≈\;1.618 · a\)
Long chord slice (\(\small{b}\))
\(\displaystyle b=\frac{a}{Φ} \)
Short chord slice (\(\small{c}\))
\(\displaystyle c=\frac{a}{Φ^2} \;\;\;\; = \frac{b}{Φ}\)
Area (\(\small{A}\))
\(\displaystyle A=\frac{1}{2} · \sqrt{25-10· \sqrt{5}}·a^2 \;\;\;\;≈0.812 ·a^2\)
Perimater(\(\small{P}\))
\(\displaystyle P= 10·b\)
Circumference Radius (\(\small{r}\))
\(\displaystyle r=\frac{1}{10} · \sqrt{50+10· \sqrt{5}}·a \;\;\;\;≈0.851 ·a\)
More polygons
Triangle • Square • Pentagon • Hexagon • Heptagon • Octagon • Nonagon • Decagon • Hendecagon • Dodecagon • Hexadecagon • N-Gon • Polygon ring • Concave hexagon • Axial Symmetric Pentagon • Irregular, stretched Hexagon • Irregular, stretched Octagon • Pentagram • Hexagram • Octagram • Star of Lakshmi
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