Lobachevsky's Theorem

Lobachevsky's Theorem pertains to hyperbolic geometry, where the usual rules of Euclidean geometry do not apply. It states that in a hyperbolic plane, the amount of space a triangle covers (its area) is directly related to the sum of its angles. Unlike in flat geometry, where the angle sum is always 180°, in hyperbolic geometry, the sum is less than 180°, and the difference (called the defect) increases as the triangle’s area grows. This theorem mathematically links the shape of a triangle to the space it occupies, demonstrating how hyperbolic geometry's properties differ fundamentally from those of flat space.