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I am amazed with your statement that "a recent article pointed...". I suspect his fact was known to the Greeks about 2000 years ago, and it is certainly mentioned in every serious trigonometry textbook.

On your question. The corresponding quantity for the spherical triangle even has a name: "modulus of the triangle", M. There is a symmetric expression $$M^2=\frac{1-\cos^2a-\cos^2b-\cos^2c+2\cos a\cos b\cos c}{\sin^2a\sin^2b\sin^2c},$$ where $a,b,c$ are the sides.

Modulus plays a role in various theorems. For example, If the modulus is greater than $1$, then either all 3 sides are greater than 90 degrees, or exactly one of the sides is less than 90 degrees.

Modulus is equal to the ratio of absolute values of two determinants: $M=|\delta/\Delta|$ where $\delta$ is the determinant of the three unit vectors pointing from the center of the unit sphere to the vertices, and $\Delta$ is the determinant of the three unit vectors which are pointing to the vertices of the polar (dual) triangle.

For this fact, see M. Berger, Geometrie, vol. 2. (The relatively modern book covering the subject). The most comprehensive book in English is W. Cahuvenet, A treatease on Plane and Spherical trigonometry, Philadephia 1850. (Poincare once said that this book contains everything one may want to know on the subject:-)

EDIT. Just found more information on this quantity. Will just give a reference: Study, Spharische Trigonometrie, orthogonale Substitutionen und elliptische Funktionen, Leipzig, 1893.

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I am amazed with your statement that "a recent article pointed...". I suspect his fact was known to the Greeks about 2000 years ago, and it is certainly mentioned in every serious trigonometry textbook.

On your question. The corresponding quantity for the spherical triangle even has a name: "modulus of the triangle", M. There is a symmetric expression $$M^2=\frac{1-\cos^2a-\cos^2b-\cos^2c+2\cos a\cos b\cos c}{\sin^2a\sin^2b\sin^2c},$$ where $a,b,c$ are the sides.

Modulus plays a role in various theorems. For example, If the modulus is greater than $1$, then either all 3 sides are greater than 90 degrees, or exactly one of the sides is less than 90 degrees.

Modulus is equal to the ratio of absolute values of two determinants: $M=|\delta/\Delta|$ where $\delta$ is the determinant of the three unit vectors pointing from the center of the unit sphere to the vertices, and $\Delta$ is the determinant of the three unit vectors which are pointing to the vertices of the polar (dual) triangle.

For this fact, see M. Berger, Geometrie, vol. 2. (The relatively modern book covering the subject). The most comprehensive book in English is W. Cahuvenet, A treatease on Plane and Spherical trigonometry, Philadephia 1850. (Poincare once said that this book contains everything one may want to know on the subject:-)