Geodesic Triangle: Calculating Area on a Sphere

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An equilateral geodesic triangle is right-angled and covers 1/8 of a sphere's surface area. The area can be calculated using the formula (1/2)πr² for a sphere of radius r. Alternatively, Girard's theorem provides another method, stating that the area of a spherical triangle is r²e, where e represents the excess angle (the sum of the triangle's angles minus π). Understanding these calculations is essential for accurately determining the area of geodesic triangles on a sphere. The discussion emphasizes the relationship between geometry and spherical surfaces.
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An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)\pir^2. But how is that obtained?
 
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Draw one on a sphere, and it'll cover 1/8 of the surface.

Or you could use Girard's theorem: the area of a spherical triangle is r2e, where e is the excess angle of the triangle (the sum of the angles, minus pi).
 

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