Sedrakyan's Area Formula: A Breakthrough on a 2,000-Year-Old Problem of Areas

Abstract: This work fills a 2,000-year-old gap and establishes a structural “missing link” in polygon geometry. Heron’s formula dates back to approximately 60 AD, while Bretschneider’s formula dates to 1842. Yet no comparable unified formula existed for polygons with more than four sides. This work shows that these two formulas, which appeared fundamentally different for centuries, are particular cases of a single general formula. The remarkable achievement is not merely that the formula uses only side lengths and certain distance lengths, but that one unified formula computes the area of any convex n-gon for every n ≥ 3. This is the key novelty, addressing a problem previously considered impossible to resolve in a similarly general way for n ≥ 5. The formula thus overcomes the perceived rigidity-related barrier to determining polygon areas for n ≥ 5. The area of a convex n-gon A₁A₂...Aₙ, where n ≥ 3, is given by

S = (1/4) × √[4(A₁A₃)²(A₂B₂)² − (a₁² − a₂² + ⋯ + (−1)ⁿ⁻¹aₙ²)²],

where AᵢAᵢ₊₁ = aᵢ, with indices taken modulo n, and B₂ is the geometrically constructed point defined in the main results. The point B₂ can be viewed as a representative point associated with the figure’s area distribution, analogous in spirit to the center of mass of a three-dimensional body. Potential applications include distance geometry, knot theory, drone networks, robotics (no GPS, no coordinates, only pairwise distances, Area is mission-critical), and other fields. For the full theorem, geometric construction, and proof, please visit: www.sedrakyans.org/sedrakyans-area-formula

mathematical physicsalgebraic topologycombinatoricsgeometric topologyrings and algebrasrepresentation theory

Audience: researchers in the topic

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Knots, graphs and groups

Series comments: Zoom - Meeting ID: 818 6674 5751 Passcode: 141592

Link: us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09

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