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Pythagoras' Theorem
Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) ... ... and you made a square on each of the three sides, then ...
... the biggest square had the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a2 + b2 = c2

Note: • • c is the longest side of the triangle a and b are the other two sides

Definition
The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Sure ... ?
Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same:

32 + 42 = 52
Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?
If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?
Write it down as an equation:

a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle. a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169 c = √169 c = 13
You can also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle. a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225
Take 81 from both sides:

b2 = 144 b = √144 b = 12 Example: What is the diagonal distance across a square of size 1?

a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2 2 = c2 c2 = 2 c = √2 = 1.4142...
It works the other way around, too: when the three sides of a triangle make c2, then the triangle is right angled.

a2 + b2 =

Example: Does this triangle have a Right Angle?

Does

a2 + b2 = c2 ?
• • a2 + b2 = 102 + 242 = 100 + 576 = 676 c2 = 262 = 676

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle? Does 82 + 152 = 162 ?
• • 82 + 152 = 64 + 225 = 289, but 162 = 256

So, NO, it does not have a Right Angle Example: Does this triangle have a Right Angle?

Does

a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ? Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !
Get paper pen and scissors, then using the following animation as a guide:

• • • •

Draw a right angled triangle on the paper, leaving plenty of space. Draw a square along the hypotenuse (the longest side) Draw the same sized square on the other side of the hypotenuse Draw lines as shown on the animation, like this:

• • • Cut out the shapes Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof
Here is one of the oldest proofs that the square on the long side has the same area as the other squares.
Watch the animation, and pay attention when the triangles start sliding around. You may want to watch the animation a few times to understand what is happening. The purple triangle is the important one.

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !

Question 1 Question 2 Question 3 Question 4 Question 5Question 6 Questi on 7 Question 8 Question 9 Question 10 Activity: Pythagoras' Theorem Activity: A Walk in the Desert
Right Angled TrianglesTrianglesProof that a Triangle has 180°Pythagorean TriplesPythagorean Theorem Algebra Proof

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S EA R C H M A TH W
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Algebra Geometry Applied Mathematics > Plane Geometry > Triangles > Triangle Properties > Recreational Mathematics > Mathematics in the Arts > Mathematics in Films > The Wizard of Oz (1939) > Calculus and Recreational Mathematics > Mathematics in the Arts > Mathematics in Television > NUMB3RS > Analysis Discrete Mathematics More... Foundations of Mathematics Geometry

Pythagorean Theorem

Pythagorean Theorem History and Terminology
Number Theory Probability and Statistics Recreational Mathematics Topology

Plotting Pythagorean Triples

Pythagorean Analogs Similar Triangles

Alphabetical Index

Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to Send a Message to the Team MathWorld Book 13,071 entries Last updated: Sat Jul 23 2011 Euclid's Proof of the

An Intuitive Proof of th

Pythagorean Theorem

Pythagorean Theorem

For a right triangle with legs

and

and hypotenuse ,

(1) Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a planetriangle to a trirectangular tetrahedron, in which case it is known as de Gua's theorem. The various proofs of the Pythagorean theorem all seem to require application of some version or consequence of the parallel postulate: proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs by similarity require the existence of non-congruent similar triangles, and so on (S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem. After receiving his brains from the wizard in the 1939 film The Wizard of Oz, the Scarecrow recites the following mangled (and incorrect) form of the Pythagorean theorem, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." In the fifth season of the television program The Simpsons, Homer J. Simpson repeats the Scarecrow's line (Pickover 2002, p. 341). In the Season 2 episode "Obsession" (2006) of the television crime drama NUMB3RS, Charlie's equations while discussing a basketball hoop include the formula for the Pythagorean theorem.
Other Wolfram Web Resources »

A clever proof by dissection which reassembles two small squares into one larger one was given by the Arabian mathematician Thabit ibn Kurrah (Ogilvy 1994, Frederickson 1997).

Another proof by dissection is due to Perigal (left figure; Pergial 1873; Dudeney 1958; Madachy 1979; Steinhaus 1999, pp. 4-5; Ball and Coxeter 1987). A related proof is accomplished using the above figure at right, in which the area of the large square is four times the area of one of the triangles plus the area of the interior square. From the figure, , so

(2)

(3)

(4)

(5)

(6)

The Indian mathematician Bhaskara constructed a proof using the above figure, and another beautiful dissection proof is shown below (Gardner 1984, p. 154).

(7)

(8)

(9)

Several beautiful and intuitive proofs by shearing exist (Gardner 1984, pp. 155156; Project Mathematics!). Perhaps the most famous proof of all times is Euclid's geometric proof (Tropfke 1921ab; Tietze 1965, p. 19), although it is neither the simplest nor the most obvious. Euclid's proof used the figure below, which is sometimes known variously as the bride's chair, peacock tail, or windmill. The philosopher Schopenhauer has described this proof as a "brilliant piece of perversity" (Schopenhauer 1977; Gardner 1984, p. 153).

Let and so

be a right triangle, . The triangles

, and

, and

be squares,

are equivalent except for rotation,

(10) Shearing these triangles gives two more equivalent triangles

(11) Therefore,

(12) Similarly,

(13) so

(14)

Heron proved that

,

, and

intersect in a point (Dunham 1990, pp. 48-53).

Heron's formula for the area of the triangle, contains the Pythagorean theorem implicitly. Using the form

(15)

and equating to the area

(16)

gives

(17)

Rearranging and simplifying gives

(18)

the Pythagorean theorem, where (Dunham 1990, pp. 128-129).

is the area of a triangle with sides , , and

A novel proof using a trapezoid was discovered by James Garfield (1876), later president of the United States, while serving in the House of Representatives (Gardner 1984, pp. 155 and 161; Pappas 1989, pp. 200-201; Bogomolny).

(19)

(20)

(21)

Rearranging,

(22)

(23)

(24) An algebraic proof (which would not have been accepted by the Greeks) uses the Euler formula. Let the sides of a triangle be , , and , and the perpendicular legs of right

triangle be aligned along the real and imaginary axes. Then

(25) Taking the complex conjugate gives

(26) Multiplying (25) by (26) gives

(27) (Machover 1996).

Another algebraic proof proceeds by similarity. It is a property of right triangles, such as the one shown in the above left figure, that the right triangle with sides , , and (small

triangle in the left figure; reproduced in the right figure) is similar to the right trianglewith sides , , and Letting (large triangle in the left figure; reproduced in the middle figure). in the above left figure then gives

(28)

(29)

so

(30)

(31)

and

(32) (Gardner 1984, p. 155 and 157). Because this proof depends on proportions of potentially irrational numbers and cannot be translated directly into a geometric construction, it was not considered valid by Euclid.
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SEE ALSO: Bride's Chair, Cosines Law, de Gua's Theorem, Peacock Tail, Pythagoras's

Theorem, Pythagorean Triple, Right Triangle, Windmill
Bottom of Form

REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 87-88, 1987. Bogomolny, A. "Pythagorean Theorem." http://www.cut-the-knot.org/pythagoras/index.shtml. Brodie, S. E. "The Pythagorean Theorem Is Equivalent to the Parallel Postulate." http://cut-theknot.org/triangle/pythpar/PTimpliesPP.html. Dixon, R. "The Theorem of Pythagoras." §4.1 in Mathographics. New York: Dover, pp. 92-95, 1991. Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 32, 1958. Dunham, W. "Euclid's Proof of the Pythagorean Theorem." Ch. 2 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990. Frederickson, G. Dissections: Plane and Fancy. New York: Cambridge University Press, pp. 28-29, 1997. Friedrichs, K. O. From Pythagoras to Einstein. Washington, DC: Math. Assoc. Amer., 1965. Gardner, M. "The Pythagorean Theorem." Ch. 16 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 152-162, 1984. Garfield, J. A. "Pons Asinorum." New England J. Educ. 3, 161, 1876. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Loomis, E. S. The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 1968. Machover, M. "Euler's Theorem Implies the Pythagorean Proposition." Amer. Math. Monthly 103, 351, 1996.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 17, 1979. Ogilvy, C. S. Excursions in Mathematics. New York: Dover, p. 52, 1994. Pappas, T. "The Pythagorean Theorem," "A Twist to the Pythagorean Theorem," and "The Pythagorean Theorem and President Garfield." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 4, 30, and 200-201, 1989.

Parthasarathy, K. R. "An -Dimensional Pythagoras Theorem." Math. Scientist 3, 137-140, 1978. Perigal, H. "On Geometric Dissections and Transformations." Messenger Math. 2, 103-106, 1873. Pickover, C. A. "The Scarecrow Formula." Ch. 103 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 217-218 and 341, 2002. Project Mathematics. "The Theorem of Pythagoras." Videotape. http://www.projectmathematics.com/pythag.htm. Schopenhauer, A. The World as Will and Idea, 3 vols. New York: AMS Press, 1977. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 123-127, 1993. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

Talbot, R. F. "Generalizations of Pythagoras' Theorem in 1987.

Dimensions." Math. Scientist 12, 117-121,

Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, p. 19, 1965. Tropfke, J. Geschichte der Elementar-Mathematik, Band 1. Berlin: p. 97, 1921a. Tropfke, J. Geschichte der Elementar-Mathematik, Band 4. Berlin: pp. 135-136, 1921b. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 202-207, 1991. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 3, 65-67, 110-113, 169-171, and 299-300, 1896. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 4, 11-12, 79-81, 168-170, 250-251, and 267-269, 1897. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 5, 73-74, 1898. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 6, 33-34 and 69-71, 1899.

CITE THIS AS:
Weisstein, Eric W. "Pythagorean Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanTheorem.html

Top of Form

S EA R C H M A TH W
Bottom of Form

Algebra Geometry > Plane Geometry > Triangles > Triangle Properties > Applied Mathematics Recreational Mathematics > Mathematics in the Arts > Mathematics in Films > The Wizard of Oz Calculus and Analysis> (1939) Recreational Mathematics > Mathematics in the Arts > Mathematics in Television > NUMB3RS > Discrete Mathematics More... Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Pythagorean Theorem

Plotting Pythagorean

Pythagorean Theorem

Triples

Pythagorean Analogs for Similar Triangles

Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom An Intuitive Proof of the Pythagorean Theorem

About MathWorld Contribute to Send a Message to the Team MathWorld Book 13,071 entries Last updated: Sat Jul 23 2011 Euclid's Proof of the Pythagorean Theorem

For a right triangle with legs

and

and hypotenuse ,

(1) Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a planetriangle to a trirectangular tetrahedron, in which case it is known as de Gua's theorem. The various proofs of the Pythagorean theorem all seem to require application of some version or consequence of the parallel postulate: proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs by similarity require the existence of non-congruent similar triangles, and so on (S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem. After receiving his brains from the wizard in the 1939 film The Wizard of Oz, the Scarecrow recites the following mangled (and incorrect) form of the Pythagorean theorem, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." In the fifth season of the television program The Simpsons, Homer J. Simpson repeats the Scarecrow's line (Pickover 2002, p. 341). In the Season 2 episode "Obsession" (2006) of the television crime drama NUMB3RS, Charlie's equations while discussing a basketball hoop include the formula for the Pythagorean theorem.

Other Wolfram Web Resources »

A clever proof by dissection which reassembles two small squares into one larger one was given by the Arabian mathematician Thabit ibn Kurrah (Ogilvy 1994, Frederickson 1997).

Another proof by dissection is due to Perigal (left figure; Pergial 1873; Dudeney 1958; Madachy 1979; Steinhaus 1999, pp. 4-5; Ball and Coxeter 1987). A related proof is accomplished using the above figure at right, in which the area of the large square is four times the area of one of the triangles plus the area of the interior square. From the figure, , so

(2)

(3)

(4)

(5)

(6)

The Indian mathematician Bhaskara constructed a proof using the above figure, and another beautiful dissection proof is shown below (Gardner 1984, p. 154).

(7)

(8)

(9)

Several beautiful and intuitive proofs by shearing exist (Gardner 1984, pp. 155156; Project Mathematics!). Perhaps the most famous proof of all times is Euclid's geometric proof (Tropfke 1921ab; Tietze 1965, p. 19), although it is neither the simplest nor the most obvious. Euclid's proof used the figure below, which is sometimes known variously as the bride's chair, peacock tail, or windmill. The philosopher Schopenhauer has described this proof as a "brilliant piece of perversity" (Schopenhauer 1977; Gardner 1984, p. 153).

Let

be a right triangle, . The triangles

, and

, and

be are equivalent

squares, and except for rotation, so

(10) Shearing these triangles gives two more equivalent triangles

(11) Therefore,

(12) Similarly,

(13) so

(14)

Heron proved that 53).

,

, and

intersect in a point (Dunham 1990, pp. 48-

Heron's formula for the area of the triangle, contains the Pythagorean theorem implicitly. Using the form

(15)

and equating to the area

(16)

gives (17 ) Rearranging and simplifying gives

(18)

the Pythagorean theorem, where (Dunham 1990, pp. 128-129).

is the area of a triangle with sides , , and

A novel proof using a trapezoid was discovered by James Garfield (1876), later president of the United States, while serving in the House of Representatives (Gardner 1984, pp. 155 and 161; Pappas 1989, pp. 200-201; Bogomolny).

(19)

(20)

(21)

Rearranging,

(22)

(23)

(24) An algebraic proof (which would not have been accepted by the Greeks) uses the Euler formula. Let the sides of a triangle be , , and , and the perpendicular legs of right triangle be aligned along the real and imaginary axes. Then

(25) Taking the complex conjugate gives

(26) Multiplying (25) by (26) gives

(27) (Machover 1996).

Another algebraic proof proceeds by similarity. It is a property of right triangles, such as the one shown in the above left figure, that the right triangle with sides , , and (small triangle in the left figure; reproduced in the right figure) is similar (large triangle in the left figure; in the above left figure then

to the right trianglewith sides , , and reproduced in the middle figure). Letting gives

(28)

(29)

so

(30)

(31) and

(32) (Gardner 1984, p. 155 and 157). Because this proof depends on proportions of potentially irrational numbers and cannot be translated directly into a geometric construction, it was not considered valid by Euclid.
Top of Form

Tail, Pythagoras's Theorem, Pythagorean Triple, Right Triangle, Windmill
Bottom of Form

REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 87-88, 1987. Bogomolny, A. "Pythagorean Theorem." http://www.cut-the-knot.org/pythagoras/index.shtml.

Brodie, S. E. "The Pythagorean Theorem Is Equivalent to the Parallel Postulate." http://cut-theknot.org/triangle/pythpar/PTimpliesPP.html. Dixon, R. "The Theorem of Pythagoras." §4.1 in Mathographics. New York: Dover, pp. 92-95, 1991. Dudeney, H. E. Amusements in Mathematics. New York: Dover, p. 32, 1958. Dunham, W. "Euclid's Proof of the Pythagorean Theorem." Ch. 2 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990. Frederickson, G. Dissections: Plane and Fancy. New York: Cambridge University Press, pp. 2829, 1997. Friedrichs, K. O. From Pythagoras to Einstein. Washington, DC: Math. Assoc. Amer., 1965. Gardner, M. "The Pythagorean Theorem." Ch. 16 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 152-162, 1984. Garfield, J. A. "Pons Asinorum." New England J. Educ. 3, 161, 1876. Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948. Loomis, E. S. The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 1968. Machover, M. "Euler's Theorem Implies the Pythagorean Proposition." Amer. Math. Monthly 103, 351, 1996. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 17, 1979. Ogilvy, C. S. Excursions in Mathematics. New York: Dover, p. 52, 1994. Pappas, T. "The Pythagorean Theorem," "A Twist to the Pythagorean Theorem," and "The Pythagorean Theorem and President Garfield." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 4, 30, and 200-201, 1989.

Parthasarathy, K. R. "An -Dimensional Pythagoras Theorem." Math. Scientist 3, 137-140, 1978. Perigal, H. "On Geometric Dissections and Transformations." Messenger Math. 2, 103-106, 1873. Pickover, C. A. "The Scarecrow Formula." Ch. 103 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 217-218 and 341, 2002. Project Mathematics. "The Theorem of Pythagoras." Videotape. http://www.projectmathematics.com/pythag.htm. Schopenhauer, A. The World as Will and Idea, 3 vols. New York: AMS Press, 1977. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea,

pp. 123-127, 1993. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

Talbot, R. F. "Generalizations of Pythagoras' Theorem in 121, 1987.

Dimensions." Math. Scientist 12, 117-

Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, p. 19, 1965. Tropfke, J. Geschichte der Elementar-Mathematik, Band 1. Berlin: p. 97, 1921a. Tropfke, J. Geschichte der Elementar-Mathematik, Band 4. Berlin: pp. 135-136, 1921b. Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 202207, 1991. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 3, 65-67, 110-113, 169-171, and 299-300, 1896. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 4, 11-12, 79-81, 168-170, 250-251, and 267-269, 1897. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 5, 73-74, 1898. Yancey, B. F. and Calderhead, J. A. "New and Old Proofs of the Pythagorean Theorem." Amer. Math. Monthly 6, 33-34 and 69-71, 1899.

CITE THIS AS:
Weisstein, Eric W. "Pythagorean Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanTheorem.html

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