Spherical Triangle: c< a+b | Tips & Examples

Thread starter prinsinn
Start date Nov 25, 2007
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Spherical Triangle

In summary, a spherical triangle is a type of triangle formed on the surface of a sphere with three points and three curved sides. It differs from a regular triangle in that its angles can add up to more than 180 degrees and its sides are related by the law of cosines. The inequality c < a + b applies to all spherical triangles and an example of this is a triangle on the Earth's surface with vertices at the North Pole, equator, and a point 90 degrees east of the North Pole.

Nov 25, 2007

prinsinn

In a spherical triangle, c< a+b

I have to talk about this subject in an oral exam for about ten minutes. I don't know how I should explain it.
If you know about a good example too, related to this subject of cource, I would really appreciate it.

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Nov 27, 2007

Shooting Star

Homework Helper

See if these are helpful.

http://mathworld.wolfram.com/SphericalTriangle.html
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node5.html

Last edited by a moderator: May 3, 2017

FAQ: Spherical Triangle: c< a+b | Tips & Examples

What is a spherical triangle?

A spherical triangle is a type of triangle that is formed on the surface of a sphere, rather than on a flat plane. It is made up of three points, or vertices, on the surface of the sphere and three sides connecting these points. The angles of a spherical triangle are measured along the surface of the sphere, and can be greater than 180 degrees.

How is a spherical triangle different from a regular triangle?

A regular triangle, or a triangle on a flat plane, has three angles that add up to 180 degrees. In contrast, the angles of a spherical triangle are measured along the surface of a sphere and can add up to more than 180 degrees. Additionally, the sides of a spherical triangle are arcs, or curved lines, rather than straight lines.

What is the relationship between the sides and angles of a spherical triangle?

In a spherical triangle, the sides are related to the angles by the law of cosines, which states that the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides multiplied by the cosine of the angle opposite the first side. This relationship only holds true for triangles on a sphere, as the sides and angles of a regular triangle on a flat plane are related by the Pythagorean theorem.

How does the inequality c < a + b apply to spherical triangles?

The inequality c < a + b is known as the triangle inequality and holds true for all triangles on a sphere. It states that the length of one side of a spherical triangle must be less than the sum of the lengths of the other two sides. This is because the shortest distance between two points on a sphere is along the arc of a great circle, and the sum of the lengths of two sides must be greater than the length of the third side in order to form a closed triangle.

Can you provide an example of a spherical triangle where c < a + b?

One example of a spherical triangle where c < a + b is a triangle on the Earth's surface with vertices at the North Pole, the equator, and a point on the equator 90 degrees east of the North Pole. In this triangle, the side connecting the North Pole to the equator (c) is shorter than the sum of the two sides connecting the equator to the point 90 degrees east of the North Pole (a + b).