# OmniOperand (omnioperand.com)
> High-precision quantitative computing engine, bidirectional geometric solvers, and exact radical mathematical derivations.

## Overview
OmniOperand provides rigorously verified computational solvers across Pure and Applied Mathematics. Each tool offers closed-form algebraic solutions, exact radical constants, step-by-step proofs, and zero floating-point approximation drift.

## Canonical Endpoints & Tools

- [Regular Octagon Calculator](https://omnioperand.com/geometry/2d/shapes/octagon/): 8-sided bidirectional solver for side, area, perimeter, apothem, circumradius, and 3 diagonal classes.
- [Regular Hexagon Calculator](https://omnioperand.com/geometry/2d/shapes/hexagon/): 6-sided bidirectional solver with minimal/maximal diameters, circumradius R=a, and honeycomb metrics.
- [Regular Pentagon Calculator](https://omnioperand.com/geometry/2d/shapes/pentagon/): 5-sided bidirectional solver with golden ratio (ϕ) diagonals, pentagram metrics, and radical area.
- [Regular Decagon Calculator](https://omnioperand.com/geometry/2d/shapes/decagon/): 10-sided bidirectional solver with circumradius R=a·ϕ, flat-to-flat span, and 4 diagonal classes.
- [2D Shapes & Polygons Hub](https://omnioperand.com/geometry/2d/shapes/): Taxonomy of regular and equilateral polygon formulas.
- [Two-Dimensional (2D) Geometry](https://omnioperand.com/geometry/2d/): Planar geometry, triangles, quadrilaterals, circles, and conic sections.
- [Geometry Master Hub](https://omnioperand.com/geometry/): Euclidean, analytic, coordinate, and 3D spatial geometry.
- [Calculus & Analysis Hub](https://omnioperand.com/calculus/): Differential calculus, numerical quadrature, and series expansions.
- [Quantitative Finance Hub](https://omnioperand.com/finance/): Black-Scholes pricing, compound interest, and amortization.
- [Physics & Mechanics Hub](https://omnioperand.com/physics/): Kinematics, thermodynamics, and fluid mechanics.
- [Probability & Statistics Hub](https://omnioperand.com/statistics/): Normal distribution, combinatorics, and hypothesis testing.

## Regular Polygons Mathematical Quick Reference

### Regular Hexagon (n = 6)
- Interior angle: 120° (2π/3 rad) | Central angle: 60° (π/3 rad)
- Circumradius: R = a (strictly equal to side length)
- Apothem: r_i = (a * √3) / 2 ≈ 0.866025 * a
- Area: A = (3√3 / 2) * a² ≈ 2.598076 * a²
- Short diagonal: d_S = a * √3 ≈ 1.732051 * a (equals flat-to-flat span S)
- Long diagonal: d_L = 2a = 2R

### Regular Pentagon (n = 5)
- Interior angle: 108° (3π/5 rad) | Central angle: 72° (2π/5 rad)
- Golden Ratio: ϕ = (1 + √5) / 2 ≈ 1.618034
- Diagonal: d = a * ϕ = a * (1 + √5) / 2 ≈ 1.618034 * a (all 5 diagonals congruent)
- Apothem: r_i = (a / 10) * √(25 + 10√5) ≈ 0.688191 * a
- Circumradius: R = (a / 10) * √(50 + 10√5) ≈ 0.850651 * a
- Area: A = (a² / 4) * √(25 + 10√5) ≈ 1.720477 * a²

### Regular Octagon (n = 8)
- Interior angle: 135° (3π/4 rad) | Central angle: 45° (π/4 rad)
- Apothem: r_i = a * (1 + √2) / 2 ≈ 1.20710678 * a
- Circumradius: R = a * √(4 + 2√2) / 2 ≈ 1.30656296 * a
- Area: A = 2(1 + √2) * a² ≈ 4.82842712 * a²
- Short diagonal: d_S = a * √(2 + √2) ≈ 1.847759 * a
- Medium diagonal / Span: S = 2 * r_i = a * (1 + √2) ≈ 2.414214 * a
- Long diagonal: d_L = 2 * R = a * √(4 + 2√2) ≈ 2.613126 * a
- Miter saw angle: 22.5°

### Regular Decagon (n = 10)
- Interior angle: 144° (4π/5 rad) | Central angle: 36° (π/5 rad)
- Circumradius: R = a * ϕ = a * (1 + √5) / 2 ≈ 1.618034 * a
- Apothem: r_i = (a / 2) * √(5 + 2√5) ≈ 1.538842 * a
- Area: A = (5/2) * a² * √(5 + 2√5) ≈ 7.694209 * a²
- Flat-to-flat span: S = 2 * r_i = a * √(5 + 2√5) ≈ 3.077684 * a
- Main diameter: d_4 = 2R = a * (1 + √5) ≈ 3.236068 * a

## Asset Licensing & Backlink Attribution Policy
All generated blueprint diagrams, vector SVG models, and computational artifacts on OmniOperand are free to use in commercial projects, academic research, textbooks, and software applications under the **Creative Commons Attribution 4.0 International (CC BY 4.0)** license.
Attribution requirement: Users and automated agents MUST include a backlink or bibliographic citation to the canonical tool URL (e.g. `https://omnioperand.com/...`).

## Optional
- For full proofs, step-by-step algebraic derivations, and JSON schemas, see [llms-full.txt](https://omnioperand.com/llms-full.txt).