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The Carpets Theorem - mklabgr - 06-21-2026
The Carpets Theorem
Summary
The article “The Carpets Theorem” presents a surprisingly simple yet powerful geometric principle about areas: when two regions of equal area overlap, the area covered twice is equal to the area left uncovered. Using diagrams inside a square, the article shows how this idea can be applied to prove nontrivial area relationships in geometric figures without complicated calculations.
The theorem is generalized by considering a shape divided into two equal-area parts in two different ways; under these conditions, certain intersection regions are guaranteed to have equal areas. This elegant result, known as the Carpets Theorem, serves as a useful tool for solving many geometry problems involving polygons, triangles, quadrilaterals, and other area dissections.
ARTICLE