Abstract A probabilistic proof of Girard’s angle excess formula for the area of a spherical triangle emerges from the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (an outcome with probability zero), and a 2-dimensional plane.

中文翻译：

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球面过剩公式的概率证明

摘要 球面三角形面积的 Girard 角余角公式的概率证明来自于观察一个无界 3 维凸锥，在原点具有单个顶点，只有三种 2 维正交投影：a 2-具有一个顶点的多维凸锥、一个二维半平面（概率为零的结果）和一个二维平面。