Hexagram Calculator
Calculator and formulas for the six-pointed star
Hexagram Calculator
The Hexagram
A hexagram is a six-pointed star formed by two overlapping equilateral triangles. Also known as the Star of David.
Hexagram structure
The diagram shows a hexagram with all relevant parameters.
Two overlapping equilateral triangles form the six-pointed star.
What is a hexagram?
A hexagram is one of the most recognized geometric figures in the world:
- Six-pointed star: Two overlapping equilateral triangles
- Star of David: Symbol of Judaism and Israel
- 12 outer sides: Each of length b
- Perfect symmetry: 6-fold rotational and mirror symmetry
- Mathematical elegance: Based on equilateral triangles
- Universal symbol: Significant in many cultures
The triangle composition
The hexagram is formed by overlaying two equilateral triangles:
Upward triangle
- Equilateral triangle with side length a
- Point directed upward
- Symbolizes heaven, masculinity, fire
- Yang principle in Chinese philosophy
Downward triangle
- Equilateral triangle with side length a
- Point directed downward
- Symbolizes earth, femininity, water
- Yin principle in Chinese philosophy
Symbolism and cultural significance
The hexagram is one of the most powerful symbols in human history:
Jewish tradition
- Star of David - symbol of Judaism
- On the Israeli flag
- Protection and luck symbol
- Connection between heaven and earth
Universal symbolism
- Balance of opposites
- Union of heaven and earth
- Harmony and perfection
- Alchemical significance
Applications of the hexagram
The hexagram finds diverse practical and cultural applications:
National symbols
- Israeli flag (Star of David)
- Military insignia
- State coats of arms and emblems
- Diplomatic symbols
Architecture & design
- Synagogue architecture
- Windows and rosettes
- Decorative elements
- Garden and landscape design
Jewelry & art
- Religious jewelry
- Amulets and talismans
- Handicrafts
- Modern design objects
Science & mysticism
- Alchemical symbols
- Crystallography (snowflakes)
- Mandala meditation
- Mathematical studies
Formulas for the hexagram
Area A
Area of the entire star
Perimeter U
12 outer star points of length b
Chord length l
Diameter of the hexagram
Chord segment b
Length of outer star points
Triangle side length a
Side length of both triangles
Triangle area
Area of an equilateral triangle
Star relationship
Hexagram = 2 triangles - overlap
√3 constant
Important constant for all calculations
Calculation example for a hexagram
Given
Find: All properties of the hexagram (Star of David)
1. Calculate basic measures
Chord length and chord segment
2. Perimeter and area
Total perimeter and star area
3. Triangle analysis
Analysis of the two overlapping triangles
4. Perfect Star of David
The complete hexagram - symbol of harmony and balance
The hexagram: Geometry meets symbolism
The hexagram is far more than just a geometric figure - it is a symbol that uniquely unites mathematics, religion, culture, and mysticism. As the Star of David, it is one of the most recognized symbols in the world, yet its mathematical elegance and geometric perfection also make it a fascinating subject of geometric study.
Geometric perfection from two triangles
The mathematical beauty of the hexagram lies in its formation:
- Two equilateral triangles: Each with perfect 60° angle division
- 180° rotation: The second triangle is rotated 180°
- √3 constant: All relationships are based on √3 ≈ 1.732
- 12 outer sides: Regular star structure
- 6-fold symmetry: Rotational and mirror symmetry
- Central hexagon: A regular hexagon forms in the interior
The Star of David: Symbol of Judaism
As the Star of David, the hexagram carries deep cultural significance:
Historical development
Originally a general protective symbol, the six-pointed star became a specifically Jewish symbol in the Middle Ages and eventually the Star of David.
Modern significance
Today central on the Israeli flag, symbol of Jewish identity and the state of Israel worldwide.
Spiritual symbolism
The two triangles symbolize the connection between heaven and earth, spirit and matter, masculine and feminine.
Protection and harmony
Traditionally used as a protective amulet, it symbolizes balance, perfection, and divine protection.
Universal symbolism and mysticism
Beyond Judaism, the hexagram has universal significance:
- Alchemy: Symbol for the union of opposites (sulfur and mercury)
- Christianity: Star of Bethlehem, symbol of Christ's birth
- Islam: Khatam or seal, protective sign
- Hinduism: Shatkona, symbol of Shiva and Shakti
- Buddhism: Symbol of enlightenment and balance
- Psychology: Mandala element for wholeness and integration
Practical applications in art and design
The hexagram finds broad practical application:
Architecture and construction
Synagogue architecture, window rosettes, floor patterns, and decorative elements frequently use hexagram structures.
Jewelry and crafts
From religious amulets to modern jewelry pieces - the hexagram is a popular design element.
Flags and emblems
As Israel's national symbol and in various military and civilian badges used worldwide.
Science and nature
Crystal structures, snowflakes, and molecular arrangements often show hexagram symmetries.
Summary
The hexagram stands as a unique example of how mathematical elegance and cultural significance can mutually reinforce each other. Its geometric perfection - born from the simple overlay of two equilateral triangles - has given it a universal appeal that transcends cultures and millennia. As the Star of David, it carries the burden and dignity of Jewish history, as a mystical symbol it speaks to the universal human longing for harmony and balance. From precise mathematical formulas to deep spiritual meditation, the hexagram shows that true beauty arises when logic and emotion, science and faith, geometry and symbolism unite into a higher unity.
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