Pythagoras' theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-02-18

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Pythagorean_theorem


00:02:22 1 Pythagorean proof
00:03:26 2 Other forms of the theorem
00:05:52 3 Other proofs of the theorem
00:06:16 3.1 Proof using similar triangles
00:11:13 3.2 Euclid's proof
00:15:22 3.3 Proofs by dissection and rearrangement
00:15:45 3.4 Einstein's proof by dissection without rearrangement
00:16:11 3.5 Algebraic proofs
00:16:22 3.6 Proof using differentials
00:18:21 4 Converse
00:19:39 5 Consequences and uses of the theorem
00:25:11 5.1 Pythagorean triples
00:27:31 5.2 Incommensurable lengths
00:28:10 5.3 Complex numbers
00:29:23 5.4 Euclidean distance in various coordinate systems
00:30:53 5.5 Pythagorean trigonometric identity
00:31:10 5.6 Relation to the cross product
00:31:22 6 Generalizations
00:31:34 6.1 Similar figures on the three sides
00:32:59 6.2 Law of cosines
00:34:39 6.3 Arbitrary triangle
00:37:57 6.4 General triangles using parallelograms
00:46:48 6.5 Solid geometry
00:48:49 6.6 Inner product spaces
00:50:44 6.7 Sets of im/i-dimensional objects in in/i-dimensional space
00:50:54 6.8 Non-Euclidean geometry
00:53:33 6.8.1 Spherical geometry
00:53:49 6.8.2 Hyperbolic geometry
00:54:27 6.8.3 Very small triangles
00:54:35 6.9 Differential geometry
00:55:33 7 History
00:55:41 8 See also
00:57:02 9 Notes
00:59:03 10 References
01:00:42 11 External links
01:06:24 Inner product spaces
01:12:29 Sets of m-dimensional objects in n-dimensional space
01:15:56 Non-Euclidean geometry
01:17:27 C. The sides are then related as follows: the sum of the areas of the circles with diameters a and b equals the area of the circle with diameter c.
01:17:40 Spherical geometry
01:24:45 Hyperbolic geometry
01:27:28 Very small triangles
01:28:37 Differential geometry
01:31:19 History
01:35:57 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8947652676007127
Voice name: en-US-Wavenet-E


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

  
    
      
        
          a
          
            2
          
        
        +
        
          b
          
            2
          
        
        =
        
          c
          
            2
          
        
        ,
      
    
    {\displaystyle a^{2}+b^{2}=c^{2},}
  where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.
Although it is often argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC) as it is he who, by tradition, is credited with its first proof, although no evidence of it exists. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references ...