Big Triangle

[Dave Davison]

What is the largest triangle one can draw before its angles add up to more than 180 degrees? Perhaps if Hey Jude has responded to a recent PM I sent you may know but leave it to others to chip in please. Dave.

[HansV]

Without context that question is impossible to answer.

[DaveA]

The sum of the angles of ALL triangles will add up to 180 degrees.

So you will never be able to have a triangle with less than or more than 180 degrees as the sum of the 3 corner angle.

[StuartR]

The sum of the angles of ALL triangles will add up to 180 degrees...

That depends on the shape of the space where the triangle is drawn. You get more than 180 degrees in a space with positive curvature.

[Dave Davison]

.....does that mean it is possible to even draw a triangle with all three angels of up to 90 degree? Maybe...Dave.

[HansV]

That's what I meant by context.

For a plane triangle, the sum of the angles is always exactly 180 degrees, but for a triangle drawn on a sphere it could be larger. Imagine a triangle with the north pole of the earth as top vertex and two points on the equator as the other vertices. The two bottom angles would each be 90 degrees, and the top angle could be anything between 0 and 180 degrees. So for example, you could have an isosceles triangle with three angles of 90 degrees.

For a triangle on a hyperbolic surface, the sum of the angles is less than 180 degrees.

[Dave Davison]

For the unconvinced..........this is what in visual form the answer given by Hans looks like which somewhat boggles the mind, theoretically it seems there can be more than 180 degrees in a triangle. Cheers Dave.
ps Sorry the attachment is a WORD doc, the FastStone Capture from the video copied as just a black image so had to revert to this method.

[HansV]

The Word document also contains a black image...

Here is an image I Googled:

spherical_triangle.png

[Hey Jude]

Yes, I only see a black image as well, but I Googled and found similar images to what Hans posted

[Hey Jude]

I tried viewing it with MS Word Viewer and Open Office but both are black
I used FastStone Capture to capture Hans' and saved it as a .png and it shows up fine

Dave, I found your image (I think)

's triangle.png

[DaveA]

Dave,
Since your question was about a "Triangle' and not a "spherical_triangle" which are two different things.
You will find that a "triangle" is a flat plane in all cases, but when you refer to a "spherical_triangle" then it is not.

A non-planar triangle is a triangle which is not contained in a (flat) plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry

[AlanMiller]

The sum of the angles of ALL triangles will add up to 180 degrees...

That depends on the shape of the space where the triangle is drawn. You get more than 180 degrees in a space with positive curvature.

Kinda funny that one. I often wonder, if mythical flat beings lived on the surface of a sphere, what would be the "constants" in their lives? For instance, we use the constancy of the 180° triangle as a matter of course. They don't have that - their triangles sum to anywhere between 180° and 540°. I wonder what equivalent "constants" they would be able to use?

Alan

[Jezza]

Alan

That opens the whole discussion into something completely different, especially if the gravity varies throughout Flatland which would identify that there is a variation in density which could hint at 3 space.

In A Square's Flatland however, perfection was the equilateral triangle of 180 degrees....wouldn't it be fun if they lived on the surface of a hypercube though!!!!