Sacred Geometry & Spatial Proportions
Interactive mathematical solvers for the Golden Ratio ($\Phi \approx 1.618$), display aspect ratios, and Pythagorean right triangle theorem with real-time SVG diagrams.
Golden Ratio (Φ)
Harmonic Scales & PartitionsAspect Ratio
16:9, 4:3, 1:1, 9:16 & 21:9Pythagoras Theorem
Interactive SVG SolverGolden Ratio (Divine Proportion) Calculator
Euclidean Extreme & Mean Ratio Composition Solver (φ = 1.6180339887...)
Expanded proportion for hero containers, sidebar ratios, and major grid columns.
Condensed proportion for secondary sidebars, captions, and harmonious padding.
Golden Section Geometric Partition (1 : 1.618)
Total: 1000 px5-Step Golden Ratio Modular Scale
r = φ^n| Harmonic Step | Formula | Calculated Size | Recommended Usage |
|---|---|---|---|
| Step -2 (b / φ²) | Step | 381.97 px | Sub-caption / Micro-tag |
| Step -1 (b / φ) | Step | 618.03 px | Small text / Caption |
| Base (b) | 1.000 × b | 1000.00 px | Body Text / Baseline |
| Step +1 (b · φ) | Step | 1618.03 px | Heading H2 / Subtitle |
| Step +2 (b · φ²) | Step | 2618.03 px | Hero Title H1 / Display |
skymotion20-20).Recommended Euclidean Geometry & Proportion Texts
Hand-selected peer-reviewed literature, optical instruments, and mechanics resources verified by physics educator Traian Anghel.
The Golden Ratio: The Story of Phi
By astrophysicist Mario Livio. The definitive exploration of the divine proportion in art, nature, and cosmology.
Sacred Geometry: Philosophy and Practice
By Robert Lawlor. Pythagorean harmony, Euclidean proportioning, and geometric harmonics.
Quadrivium: Number, Geometry, Music, & Cosmology
The four classical liberal arts combining Euclidean geometry, harmonic intervals, and sacred proportions.
Using the golden ratio calculator
This tool applies the constant φ = (1 + √5)/2 to a base dimension, producing larger and smaller harmonics plus a modular scale that can be used to explore proportional layouts and typography.
Method & formula
The expanded value is bφ and the reduced value is b/φ. A golden-section division separates a whole into approximately 61.803% and 38.197%, so the whole-to-major ratio equals the major-to-minor ratio.
Inputs
Enter any positive base dimension. Although results are displayed in pixels for design convenience, φ is dimensionless and the same multiplication or division applies to centimeters, meters, points, or abstract units.
How to read the result
Treat the outputs as proportional candidates, not mandatory design rules. Round pixel values according to the rendering context and test whether the resulting hierarchy improves readability, balance, and available space.
Assumptions & limits
The golden ratio does not guarantee aesthetic quality, usability, or biological significance. Layout constraints, content length, accessibility, responsive breakpoints, and optical alignment may justify different proportions.
Reference: Euclid’s extreme and mean ratio and the modern constant φ ≈ 1.6180339887.
Display & Media Aspect Ratio Dimension Calculator
Digital Screen, Sensor & Video Composition Proportions (SMPTE & ISO)
Preserving image and display proportions
The aspect-ratio calculator finds a matching height for a chosen width while preserving the relationship between horizontal and vertical dimensions. Presets cover common video, photography, display, and social-media formats.
Method & formula
For ratio rw:rh and width W, the corresponding height is H = W × rh/rw. The decimal aspect ratio is rw/rh; scaling both ratio terms by the same factor leaves the shape unchanged.
Inputs
Select a preset or enter a custom ratio, then provide the target width. The preview visualizes orientation and shape while the numeric output supplies the proportional height.
How to read the result
A 1920-pixel width at 16:9 produces 1080 pixels in height. If the result is fractional, choose a rounding method appropriate to the export pipeline and verify that codecs or platforms do not require even-numbered dimensions.
Assumptions & limits
Maintaining aspect ratio prevents geometric stretching but does not prevent cropping, letterboxing, resolution loss, or compression artifacts. Platform specifications can change and may impose file-size, safe-area, or codec constraints beyond dimensions.
Reference: Elementary proportional geometry: W/H = rw/rh.
Pythagorean Theorem & Right-Angled Triangle Solver
Euclidean Geometry Fundamental Relationship (Euclid Elements Book I, Prop 47)
Solving a right triangle with the Pythagorean theorem
The interactive diagram calculates the missing side of a right triangle and updates its geometry as the two perpendicular legs change. It also illustrates the relationship between side lengths and the acute angles.
Method & formula
For perpendicular legs a and b and hypotenuse c, a² + b² = c². Therefore c = √(a²+b²); if the hypotenuse and one leg are known, the other leg is √(c²−a²).
Inputs
Enter positive lengths for the two legs. The units are arbitrary but must be consistent. The diagram rescales for visibility, while the displayed numerical result preserves the entered proportion.
How to read the result
The hypotenuse is always opposite the 90° angle and is the longest side. A 3–4–5 triangle is an exact integer example: 3² + 4² = 9 + 16 = 25 = 5².
Assumptions & limits
The theorem applies only to Euclidean right triangles. Measurements with uncertainty require error propagation, and map, geodesic, or very large-scale distances may need spherical or ellipsoidal geometry instead.
Reference: The Euclidean Pythagorean theorem for right triangles.
skymotion20-20).Recommended Euclidean Geometry & Proportion Texts
Hand-selected peer-reviewed literature, optical instruments, and mechanics resources verified by physics educator Traian Anghel.
The Golden Ratio: The Story of Phi
By astrophysicist Mario Livio. The definitive exploration of the divine proportion in art, nature, and cosmology.
Sacred Geometry: Philosophy and Practice
By Robert Lawlor. Pythagorean harmony, Euclidean proportioning, and geometric harmonics.
Quadrivium: Number, Geometry, Music, & Cosmology
The four classical liberal arts combining Euclidean geometry, harmonic intervals, and sacred proportions.
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