Polygons & Geometric Shapes
High-precision calculators for equilateral and regular polygons. Each solver is equipped with bidirectional solving across all dimensions, live Cartesian SVG blueprints, exact radical algebraic proofs, and machine-readable LLM prompts.
Regular Octagon
8-sided regular polytope. Bidirectional solver for side length, area, inradius, circumradius, span, and all 3 diagonal classes with exact radical derivations.
Regular Hexagon
Equiangular polygon with 6 equal sides. Exact formulas for minimal/maximal diameters, apothem r = a\sqrt{3}/2, and honeycomb tessellations.
Regular Pentagon
5-sided polygon governed by the golden ratio \phi = \frac{1+\sqrt{5}}{2}. Diagonals form the classic self-intersecting pentagram star.
Regular Decagon
10-sided regular polygon with interior angles of 144°. Inradius and circumradius relationships expressible through radical expressions of \sqrt{5}.
For any regular polygon with n sides of length a:
- Interior Angle: \alpha = \frac{(n - 2) \times 180^\circ}{n} = \pi \left(1 - \frac{2}{n}\right) \text{ rad}
- Apothem (Inradius): r_i = \frac{a}{2 \tan(\pi / n)}
- Circumradius: R = \frac{a}{2 \sin(\pi / n)}
- Total Area: A = \frac{n a^2}{4 \tan(\pi / n)} = \frac{1}{2} n R^2 \sin\left(\frac{2\pi}{n}\right)
- Total Diagonals: N_d = \frac{n(n - 3)}{2}