Spherical Triangle: Sum of Two Sides is Always Bigger than Third Side

In summary, a spherical triangle is a triangle drawn on the surface of a sphere using three arcs of great circles. The sum of the angles in a spherical triangle is always greater than 180 degrees and less than 540 degrees, and the sum of two sides is always greater than the third side due to the shortest distance between two points on a sphere being along a great circle. This rule does not apply to triangles on a flat surface, where the sum of two sides can be equal to or smaller than the third side. The relationship between the sum of two sides and the third side in a spherical triangle can vary based on the size and shape of the triangle, with larger triangles having a greater difference between the two sums.
  • #1
prinsinn
10
0
Do you know any good example (or proof) that shows that the sum of two sides is alwasys bigger than the third side in a spherical triangle.
 
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  • #2
spherical "law of cosines"
 
  • #3
How about simply "a straight line is the shortest distance between two points"?
 
  • #4
Think about the converse. What is a triangle like when the third side is equal to or less then the 2 other sides put together?
 

FAQ: Spherical Triangle: Sum of Two Sides is Always Bigger than Third Side

What is a spherical triangle?

A spherical triangle is a triangle drawn on the surface of a sphere. It is formed by three arcs of great circles, which are the largest circles that can be drawn on a sphere.

What is the sum of the angles in a spherical triangle?

The sum of the angles in a spherical triangle is always greater than 180 degrees, and can vary depending on the size and shape of the triangle. However, the sum will always be less than 540 degrees.

Why is the sum of two sides always bigger than the third side in a spherical triangle?

This is due to the fact that a spherical triangle is formed by three arcs of great circles, which are the shortest distance between two points on a sphere. The shortest distance between two points on a sphere is always along a great circle, making the sum of the two sides longer than the third side.

Does this rule also apply to triangles on a flat surface?

No, the rule only applies to spherical triangles. In a flat surface, the sum of the two sides can be equal to, or even smaller than, the third side.

How is the sum of two sides related to the third side in a spherical triangle?

The sum of two sides in a spherical triangle is always greater than the third side, but the exact relationship can vary based on the size and shape of the triangle. In general, the larger the triangle, the greater the difference between the sum of the two sides and the third side.

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