Area of triangle on a sphere (not spherical triangle)

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How do I find the area of a triangle on a sphere, and the triangle is not a spherical triangle, for example, the triangle is formed with two geodesics and a line of latitude.

Is there a specific name for this kind of triangle?

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You should compute surface integral I suppose. In such case I'd change variables to those corresponding to three planes that cutting your 'triangle' out of sphere.