Enter known sides and angles to solve a spherical triangle on a sphere
Side a (deg)
Side b (deg)
Side c (deg)
Angle A (deg)
Angle B (deg)
Angle C (deg)
Enter at least 3 values. SSS uses spherical law of cosines.
Result
a
-
b
-
c
-
Angle A
-
Angle B
-
Angle C
-
Detailed Derivation
Spherical Law of Cosines
cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A)
cos(b) = cos(a)cos(c) + sin(a)sin(c)cos(B)
cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)
Spherical Excess E = A + B + C - 180
A spherical triangle is formed by three great circle arcs on a sphere. Its sides and angles satisfy the spherical law of cosines, which differs from the planar version. The angle sum always exceeds 180 degrees.
⚠Sides are measured in angular degrees (central angles). For Earth, 1 degree of arc = 111.32 km. Enter sides as angular measures.
What Is a Spherical Triangle?
A spherical triangle is formed by three arcs of great circles on a sphere. It is fundamental to navigation, astronomy, and geodesy. Unlike planar triangles, spherical triangles have angle sums exceeding 180 degrees and side lengths measured as angles.
Great Circles
Spherical triangle sides are segments of great circles - the intersection of the sphere with a plane through the sphere center.
Angle Sum
Sum > 180 degrees. The spherical excess E = sum - 180 is proportional to the triangle area on the sphere.
Law of Cosines
cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A). This is the main tool for solving spherical triangles.
Navigation
Used to compute great circle distances and bearings between two points on Earth using known latitudes and longitudes.
Teaching Example: SSS case: a=60, b=70. Find angle A using spherical law of cosines: cos(A) = [cos(a)-cos(b)cos(c)]/[sin(b)sin(c)]. For a right spherical triangle with C=90: cos(c) = cos(a)cos(b).
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