# 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/): Bidirectional regular octagon solver. Given any one known parameter (side length, area, perimeter, inradius/apothem, circumradius, short diagonal, medium diagonal/span, or long diagonal), it computes all remaining 7 parameters with machine precision and exact radical formulas.
- [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 Octagon Mathematical Quick Reference
- Interior angle: 135° (3π/4 rad)
- Exterior/Central angle: 45° (π/4 rad)
- Half-central angle: 22.5° (π/8 rad)
- Apothem (Inradius): r_i = a * (1 + √2) / 2 ≈ 1.20710678 * a
- Circumradius: R = a * √(4 + 2√2) / 2 ≈ 1.30656296 * a
- Area from side: A = 2(1 + √2) * a² ≈ 4.82842712 * a²
- Side from area: a = √(A / [2(1 + √2)]) = √(A * (√2 - 1) / 2) ≈ 0.45508986 * √A
- Short diagonal (across 2 sides): d_S = a * √(2 + √2) ≈ 1.84775907 * a
- Medium diagonal (Span / Height flat-to-flat): S = 2 * r_i = a * (1 + √2) ≈ 2.41421356 * a
- Long diagonal (across opposite vertices): d_L = 2 * R = a * √(4 + 2√2) ≈ 2.61312593 * a
- Carpentry corner cut from span S: Corner cut leg = (S - a) / 2 = a / √2 ≈ 0.70710678 * a
- Miter saw angle: 22.5°

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