[Mathematics] Sum of angles of triangle [Updated]

[patbob]

d@nish wrote:

I guess it was non Euclidean or something geometr

Spherical geometry. A fun topic that leads to all sorts of amusing party games... like how to make someone rotate their arm without rotating their arm :)

patbob

[Stefan_Lang]

Try this exercise to get a better understanding what this thing about angles not adding up to 180 degrees is about: 1. Take a sheet of paper. Draw an arbitrary triangle on it. Measure the angles and add up. Within a reasonable tolerance, you should get about 180 degrees. 2. Now take this paper and roll it up to form a cylinder. When you look at the triangle, you will notice none of the angles have actually changed. If you want you can glue your cylinder together so you have both hands free to measure them again, but you won't notice any changes. So the angles still sum up to 180 degrees, although the surface this triangle is drawn upon is no longer flat! I'll come to this later. 3. Find a balloon (non-inflated) and some kind of pen that would write on it, such as CD markers. Lay the balloon flat on your desk, and use your pen and a ruler to draw a triangle on it, as big as you can. Measure the angles. This time, due to the uneven surface the sum of angles will probably slightly deviate from, yet still approximate 180 degrees well enough (I'd expect the number you get to be anywhere between 160 and 200). Now inflate it to a nice round ball and tie it up. Measure the angles again. Notice something? All angles suddenly got considerably bigger! Depending on how big your triangle is, your angular sum should now be 50-100 degrees bigger than before, maybe even more. What are the lessens learned? 1. Regardless of how you draw it, a triangle *will* have angles that sum up to approx. 180 degrees when drawn on a flat surface. 2. You may fold, roll up, crumple or otherwise deform a paper you have drawn the triangle on, the angles will not change, nor the sum of angles - this means: A surface does not actually have to be flat for this fundamental theorem of trigonometry to hold! It is still valid on many uneven surfaces! 3. Nevertheless, The theorem stops to be true on many other kinds of surfaces, such as spheres, or balloons. In fact this is true for most non-flat surfaces. The one notable exception is what you've learned in case two: a surface that you can 'wrap up' in a flat sheet of paper (such as a cylinder) is geometrically equivalent to a flat surface. You can not exactly wrap up a sphere in a sheet of paper though, which is why at this point the geometry rules change.

[AngryDane]

The 180 degrees has to do with right triangles only! Smiles

[costas0811]

ok triangles are out, and pi is not finite so circles are out. i hate shapes.

[Luc Pattyn]

Now we're talking. Don't tell me something can't be done! :)

Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles]

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[Gregory Gadow]

That would only happen if the average curvature of the space containing the triangle had zero curvature. By definition, an surface with an average curvature of zero has zero curvature, so my original statement remains correct. ;P

[Stefan_Lang]

Nope, not at all. The sum of angles in a planar triangle is 180 degrees. Always. Try it out.

[RC Roeder]

No, it means that the drawing is not truly 30, 60 and 90. Having been a draftsperson in a former life, the advent of CAD vs. hand drawing improved the accuracy. The plastic drafting triangles are approximate, very close, but never 30.00000000000000000, 60.00000000000000000 or 90.00000000000000000 degrees. You could also state, that the sum of the angles in radians is pi. I spend years developing software for AutoCAD that created parametric parts, and I find the concept of using radians for these calculations easier.

[Eusebiu Marcu]

:-O Hello... Basically when you are asking (your self or others) a geometry question you also have to state the nature of the geometry you are referring. As you know, there are a number of geometries. What we have learnt first was Euclidean Geometry that is based on a number of axioms. In this type of geometry, the sum of the angles of a triangle is always 180 degrees. But, for example, in hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees. So your statement is not always true or false. Depends on context. :) But let's return to Euclidean geometry. You say that no one can draw a line(better said a segment - a line is by definition infinite) of length sqrt(3). Before I go on, I'd like to ask, how do you measure the length? Can you draw a segment of length 1? :). Here you want to mix math with engineering. First of all, in math there are no units. I think they are in physics and fields derived from physics. Anyway, so lets say I can say that given two points we can draw a segment starting from one point and ending to the other (of course, you can ask: how would we know that the line that we draw using those points, contains the points? well, we don't from the engineering point of view. we take this as an axiom. From the math's point of view, a point is something infinite small.). So we have a segment and lets say that it has the length of 1 unit (you name it: inch, centimeter, meter, light-year... :) ). I say I can draw a segment that has the length sqrt(3) like this (using the ruler and compass): first I create a isosceles right triangle having both cathetus of length 1 starting from two points. I consider the segment given by the two starting points as a cathetus and then I draw the rest of the line. Using the compass, I draw a circle of radius 1 and center the second point (B) and get a third point (C) that is antipodal with the first point (A). Now, I draw 2 circles of center A and C with the same radius > 1. These two circles will intersect in two points D1 and D2. If I connect D1 and D2 I will pass though B. Now I will draw a circle of center B and radius 1 that will intersect the segment D1-D2 in 2 points E1 and E2. Well, the segments AE1 and AE2 will have the length sqrt(2). If I repeat the construction using the E1 as B I will obtain 2 other points G1 and G2, and AG1 = AG2 = sqrt(3). I am sure that now you will can draw any segment with length sqrt(n), n > 0. :) Edit: for those out there who

[Tio Filito]

Im agree with you, the angles of any triangle sum 180 degrees, thats math. The real problem is when we want to draw that triangles, our tools and senses has limits. Math is something theoric with theric limits and drawing is something real, with real limits, In theory we can do a lot of things, in the reality we has lot of limits. Think this, you have something with weight of 1 kg in the second floor of a building if you take that thing ando go to the fifth floor the weight will be diferent than 1, because the gravity is not the same, thats another example of the big diference between the teorical world and the real world.

[Trevortni]

I remember seeing this covered in Sphereland[^], a sequel to the classic Flatland, in which one of A. Square's descendants begins to explore Flatland, and discovers many interesting things about this world, including, by way of the very divergence of angles you're referring to, the fact that Flatland is in actuality Sphereland. However, the divergence from 180 degree angles can only be detected between objects that are very far apart - such as stars, as you mentioned. This can only be explained through curvature of space, as others have already mentioned in several subthreads. I hope this helps.

[randomusic]

By your own definition perhaps ;) Even if we would use your definition, it still would not be necessarily true. It is the corners that counts here. Whatever happens in other parts of the surface, even if looking only at "the space containing the triangle" does not affect the corner angles. So one can certainly design a surface which has an "average curvature" which is not zero, but which has an "average curvature" of zero if we only consider the three points of the corners. And that would give us a triangle with the angle sum of 180 degrees.

[Lost User]

Not only a 30, 60, 90 degree is possible, as it's commercially sold world wide. It's called a 30º square. http://image.shutterstock.com/display_pic_with_logo/87179/87179,1207723301,2/stock-vector-vector-drafting-tools-degree-triangle-degree-triangle-ruler-french-curve-and-a-11307385.jpg[^]

[virang_21]

In 2D any triangle's angles sum is going to be 180 no matter what. Put the same theory to 3D and it fails. Imagine drawing triangle on a ball. The lines are going to be curved and in that case the sum of angles of that triangle is not going to be 180. page 128 in the book "Mathematics, The Science of Patterns" by Keith Devlin, a Scientific American Library book. There was the picture of a pseudosphere and on that the angle sum of a triangle is less than 180.

[faiko]

Pretty interesting to find out. It is always true that sum of angles is always 180 degree if and only if the triangle is lied on a flat surface(plane). Likewise, however, Hyperbolic & Spherical trigonometry don't show this kind of attribute. Hyperbolic triangle can have sum of angles less than 180 degree, whereas Spherical triangle can have more than 180 degree. These would be another case of triangle whose the plane is not flat.

[Gregory Gadow]

Actually, the vertices are irrelevant. What are important are the ANGLES, as that is what we are measuring. If the angles of a triangle add up to 180 degrees, then space containing the polygon has zero curvature, by the operative definitions of zero-curving space. Space elsewhere, away from the polygon, might have a different curvature, but that is also irrelevant.

[Mark Ginnane]

This is a bit silly, really. You are equating your ability to draw a triangle with the validity of a Euclidean geometric rule! In Euclidean geometry the sum of the angles of a triangle MUST add up to 180 degrees. However, you can have other geometries where this is not true. For example, Lobachevsky created a geometry in which Euclid's 5th postulate (that for any given line and a point not on that line, there is one parallel line through the point not intersecting the line) was false, that is, there is more than one line that can be extended through any given point parallel to another line of which that point is not part. One of the consequences of this is that the sum of the angles of a triangle must be LESS than 180 degrees. This geometry was an idle curiosity for over 100 years until it was found to be useful in relativistic physics.

[gofrancesc]

Hello, you can draw this line, and you stated how: - Draw two perpendicular lines. - Take your compass and open it some width; it defines a unit. - With your compass measure, from the intersection of the two lines, measure one unit right and two units up. - Link the two resultant points with a line. This line measures SQRT(3). So you can effectively draw this line. Sum of angles of a triangle is 180 degree, yes, in an Euclidean geometry (plane geometry). But in other geometries this sum can be bigger, take this example: - Take the sphere as the surface of this geometry. - A "straight line" is the line that joins two points with the minimum length, in this case is a circle of maximum length. Over the sphere there are no paralel lines. - Consider the triangle defined by: the North Pole, the Ecuador at meridian 0, and the Ecuador at meridian 90. In this triangle all three angles measure 90 degrees. I cannot recommend you any bibliography, I'm not an expert on Geometry. But some articles about non-Euclidean Geometry in Wikipedia could be a good start in this theme. Regards, Francesc

[randomusic]

> If the angles of a triangle add up to 180 degrees, then space > containing the polygon has zero curvature, by the operative > definitions of zero-curving space. From where are you getting all these strange definitions? Are you just making them up on the fly to fit your original claim or? Perhaps you should qoute your sources, if you have any. In a flat space, the sum of the squares of the sides of a right angled triangle is equal to the square of the hypotenuse. This relationship does not hold for curved spaces. This means that if this is true for any right angled triangle in a given surface, then it is a flat space. If not, then it is a curved space. From this follows that a triangle in a curved space might have an angle sum of 180 degrees. You are making the mistake of saying that all birds are crows. When only the reverse is true, all crows are birds. Magnus

modified on Friday, April 16, 2010 7:05 AM

[gofrancesc]

Oops, of course, if one side is 1 and other side is 2 the hypotenuse measures SQRT(5), not SQRT(3). But there is not a problem; you can construct a triangle with sides 1 and SQRT(2) at the 90 degrees angle, and the hypotenuse will measure SQRT(3). Francesc