The Pythagorean Theorem: Proofs, Triples and Applications

What is the Pythagorean theorem?

In a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem states that a^2+b^2=c^2. The legs are the two sides that meet at the right angle; the hypotenuse is the side opposite it, which is always the longest (see right triangles).

The equation is about area. The square built on the hypotenuse has exactly the same area as the two squares built on the legs added together, which is the picture in the tool below. The name honors Pythagoras of Samos (about 570–495 BCE), but the relation was known in Babylonia, India and China long before him; the history is at the end.

Two things follow at once:

• A missing hypotenuse: c=√(a^2+b^2). For legs 3 and 4 it is √(9+16)=5. For two legs of 1 it is √2, which is irrational, and for legs 6 and 4 it is √(52)=2√(13) once you simplify the radical.
• A missing leg: b=√(c^2-a^2), with a subtraction under the root, because the hypotenuse is the big one. For c=13 and a=5 it is √(169-25)=12.

Keep every length in the same unit, and keep the square root as a root, or in simplified form, until the final step. The theorem needs a right angle: for other triangles see the law of cosines in the section on generalizations below.

Try it: give two sides and get the third, exactly, with a square drawn on every side.

How do you prove the Pythagorean theorem?

There are hundreds of proofs, and four of the clearest rearrange areas, use similar triangles, shear squares into rectangles, or add up a trapezoid; each takes only a few lines. Here they are, starting with the one in the tool below.

1. Rearranging four triangles. Take four copies of the right triangle, each of area \frac12ab, and fit them inside a square of side a+b. Two arrangements are possible. In the first, the triangles leave a tilted square of side c in the middle. In the second, the same triangles leave two squares, one of side a and one of side b. The big square has the same area both times, and so do the four triangles, so what is left over must be equal: c^2=(a+b)^2-4·\tfrac12ab=a^2+b^2. This is the diagram of the Chinese Zhoubi Suanjing. Slide a and b and switch the arrangement below.

2. Similar triangles. Draw the altitude from the right angle to the hypotenuse. It cuts the hypotenuse c into two pieces p and q with p+q=c, and cuts the triangle into two smaller ones, each similar to the whole by AA, as in similar triangles. Comparing the ratios of corresponding sides gives a/c=p/a and b/c=q/b, that is, a^2=cp and b^2=cq. Add them: a^2+b^2=c(p+q)=c· c=c^2.

3. Euclid's proof (Elements I.47). The same two equations say that if the altitude is extended through the square on the hypotenuse, it splits that square into two rectangles, one of area cp=a^2 and one of area cq=b^2. Euclid showed this without ratios, by shearing: a square and a rectangle on the same base and between the same parallels, cut in half by a diagonal, give triangles of equal area, so each square on a leg equals one rectangle of the big square.

4. Garfield's trapezoid (1876). Put two copies of the right triangle side by side so that the hypotenuses form a right angle; with a third right triangle of legs c and c in between, the figure is a trapezoid with parallel sides a and b and height a+b. Its area is \frac12(a+b)(a+b), and also the sum of the three triangles, \frac12ab+\frac12ab+\frac12c^2. So \frac12(a+b)^2=ab+\frac12c^2, which becomes a^2+2ab+b^2=2ab+c^2 and again a^2+b^2=c^2. James Garfield found it while he was a member of the US House of Representatives, and he became the 20th president five years later.

A collection by Elisha Loomis, in its 1940 edition, gathers about 370 proofs of the theorem.

How do you use the converse to check for a right angle?

If the three sides of a triangle satisfy a^2+b^2=c^2, the angle opposite c is a right angle; this converse is how builders square a corner with a 3–4–5 rope.

Why the converse is true. Suppose a triangle has sides a, b, c with a^2+b^2=c^2. Build a right triangle with legs a and b. By the theorem its hypotenuse is √(a^2+b^2)=c, so it has the same three sides as the given triangle. Two triangles with three equal sides are congruent (SSS), so the given triangle has a right angle too. Euclid proves it as proposition I.48, the last of Book I.

The 3–4–5 rope. Tie a loop of rope with 12 equal spaces marked by knots. Hold the knots 3 apart and 4 apart and pull the loop tight into a triangle: the sides are 3, 4 and 5, and 9+16=25, so the corner between the sides 3 and 4 is exactly square. Surveyors and builders have done this for thousands of years. Any multiple works equally well, such as 6–8–10 for a large room.

The diagonal check. A frame 2.4 m by 1.8 m is a rectangle only if its diagonals are 3.0 m, because 1.8^2+2.4^2=3.24+5.76=9.00=3.0^2. If a diagonal is longer or shorter, the frame is leaning.

What if the test fails? Then the angle is not a right angle, and the size of c^2 against a^2+b^2 tells which way it goes: a larger c^2 means an obtuse angle, a smaller one an acute angle, as set out under the types of triangles. The sides 5, 7, 9 give 25+49=74<81, so the biggest angle is obtuse.

What are Pythagorean triples and how do you find them?

A Pythagorean triple is a set of three positive whole numbers a,b,c with a^2+b^2=c^2, and Euclid's formula (m^2-n^2, 2mn, m^2+n^2) produces every primitive one. The triple (3,4,5) is the smallest; (5,12,13) and (8,15,17) follow. A multiple of a triple is a triple ((6,8,10) is twice (3,4,5)), so the interesting ones are the primitive triples, with no common factor, as in the greatest common divisor.

Euclid's formula. For whole numbers m>n>0 the triple (m^2-n^2, 2mn, m^2+n^2) works, because (m^2-n^2)^2+(2mn)^2=m^4-2m^2n^2+n^4+4m^2n^2=(m^2+n^2)^2. It is primitive exactly when m and n are coprime and one of them is even, and then it gives every primitive triple once.

There are 16 primitive triples with hypotenuse at most 100, and 52 triples in all once the multiples are counted.

Patterns in every triple. In a primitive triple exactly one leg is even and the hypotenuse is odd. Among a,b,c one leg is a multiple of 3, one leg is a multiple of 4, and one of the three numbers is a multiple of 5, so abc is always divisible by 60 (see divisibility). The area \frac12ab is always a whole number.

All triangles with a given leg. Write a^2=c^2-b^2=(c-b)(c+b) and list the ways to factor a^2 into two numbers of the same parity. For a=12, a^2=144=2·72=4·36=6·24=8·18, giving (c,b)=(37,35),(20,16),(15,9),(13,5): four right triangles with a leg of 12.

Explore the lists below, then walk the tree.

The tree of primitive triples. In 1934 the Swedish mathematician Berggren found that all primitive triples fit in one tree. The root is (3,4,5) and every triple has exactly three children, found with three fixed linear rules, so each primitive triple appears exactly once. With the odd leg first, (3,4,5) has the children (5,12,13), (21,20,29) and (15,8,17).

The triples are not just curiosities: \left(\frac35,\frac45\right) is a point of the unit circle because 3^2+4^2=5^2, and every triple gives such a point, as in the equation of a circle.

How does the theorem give distances and diagonals?

The distance between two points is the hypotenuse of a right triangle whose legs are the horizontal and the vertical differences, so d=√((x_2-x_1)^2+(y_2-y_1)^2); one more use of the theorem gives the diagonal of a box in space.

In the plane. From (1,2) to (7,10) the legs are 6 and 8, so the distance is √(36+64)=10. Comparing squared distances needs no square root at all, which is how the quadrilateral tests check equal sides, and the equation (x-h)^2+(y-k)^2=r^2 of a circle is the distance formula saying "the distance to the center is r".

In space. Use the theorem twice. The diagonal of the floor of a box l× w is √(l^2+w^2), and this diagonal and the height h are the legs of a second right triangle whose hypotenuse is the diagonal of the box: d=√(l^2+w^2+h^2). A box 2×3×6 has diagonal √(4+9+36)=7, and the longest rod that fits in a box 3×4×12 is √(9+16+144)=13. A unit cube has diagonal √3. The same pattern, a sum of squared differences under a root, gives the distance in any number of dimensions.

Everyday uses.

• A ladder 5 m long with its foot 1.4 m from the wall reaches √(25-1.96)=4.8 m up the wall.
• A shortcut across a field 60 m by 80 m is 100 m, against 140 m along the edges.
• A screen size is its diagonal. A 55-inch screen with the shape 16:9 has width and height proportional to 16 and 9, and √(16^2+9^2)=√(337), so it is about 47.9 in wide and 27.0 in high, since (55·16)/(√(337))≈47.9 and (55·9)/(√(337))≈27.0.
• A paved diagonal path across a rectangular garden 30 m by 40 m is 50 m long; at 20 dollars per meter it costs 1,000 dollars.

Can the Pythagorean theorem be generalized?

Yes: the law of cosines extends it to every triangle, any similar shapes drawn on the three sides have areas that add up in the same way, the distance formula carries it into any dimension, and for powers above two the equation has no whole-number solutions at all.

• The law of cosines. For any triangle, c^2=a^2+b^2-2ab\cos C. At a right angle \cos C=0 and the correction vanishes, which is the Pythagorean theorem; for an acute angle the correction subtracts, for an obtuse angle it adds. See solving a triangle.
• Other shapes on the sides. Area grows with the square of length, so any similar figures built on the three sides satisfy A_a+A_b=A_c: semicircles, equilateral triangles, even copies of your favorite drawing. With semicircles, Hippocrates of Chios (5th century BCE) drew the two crescents (lunes) outside the hypotenuse's semicircle and showed that, for a right triangle, the two lunes together have exactly the area of the triangle.
• Vectors and higher dimensions. If two vectors are perpendicular, |u+v|^2=|u|^2+|v|^2; this is the theorem for the triangle they span, and the distance in any number of dimensions is built from it.
• A curved surface. On a sphere of radius 1, a triangle with a right angle satisfies \cos c=\cos a\cos b instead, with the sides measured along the surface. The plain theorem needs a flat plane.
• Higher powers. The equation x^2+y^2=z^2 has infinitely many whole-number solutions, the triples above. For x^n+y^n=z^n with n\ge3 there are none in positive whole numbers. Pierre de Fermat wrote this in a margin around 1637, saying that he had a proof too long for the margin; the first proof was found by Andrew Wiles, with a gap closed by Richard Taylor, in 1995. This is Fermat's Last Theorem.

How do you use the Pythagorean theorem in code?

Use math.hypot for lengths, and integers with math.isqrt for exact tests. The square root of a float is rounded, so a right angle is best tested with whole numbers, and a huge or tiny length is best computed with hypot, which cannot overflow.

from math import hypot, dist, isqrt, sqrt, gcd

def is_triple(a, b, c):
    return a * a + b * b == c * c                 # exact for whole numbers

def hyp_exact(a, b):                              # (isqrt, True) when the hypotenuse is whole
    s = a * a + b * b
    c = isqrt(s)
    return c, c * c == s

def primitive_triples(limit):                     # Euclid's formula
    out = []
    m = 2
    while m * m + 1 <= limit:
        for n in range(1, m):
            if (m - n) % 2 and gcd(m, n) == 1 and m * m + n * n <= limit:
                a, b = sorted((m * m - n * n, 2 * m * n))
                out.append((a, b, m * m + n * n))
        m += 1
    return sorted(out, key=lambda t: t[2])

>>> is_triple(5, 12, 13), is_triple(4, 5, 6)
(True, False)
>>> hyp_exact(8, 15), hyp_exact(1, 1)
((17, True), (1, False))
>>> primitive_triples(30)
[(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29)]
>>> len(primitive_triples(100))
16
>>> hypot(3, 4), dist((1, 2), (7, 10)), hypot(2, 3, 6)   # 2D, between points, 3D
(5.0, 10.0, 7.0)
>>> sqrt(1.2 ** 2 + 3.5 ** 2) == 3.7                 # should be True: 1.2, 3.5, 3.7 is a triple
False
>>> hypot(1.2, 3.5) == 3.7
True
>>> sqrt(1e200 ** 2 + 1e200 ** 2)                    # squaring overflows
OverflowError: (34, 'Result too large')
>>> hypot(1e200, 1e200)
1.414213562373095e+200

Math.hypot(3, 4)                                   // 5
Math.hypot(1, 2, 2)                                // 3: the diagonal of a 1 by 2 by 2 box
const isTriple = (a, b, c) => a * a + b * b === c * c   // exact for whole numbers
isTriple(20, 21, 29)                               // true
Math.sqrt(1.2 ** 2 + 3.5 ** 2) === 3.7             // false: the float sum rounds, as in Python

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

bool isTriple(int a, int b, int c) { return a * a + b * b == c * c; }   // exact for whole numbers

int isqrt(int n) { int r = 0; while ((r + 1) * (r + 1) <= n) r++; return r; }

int gcd(int a, int b) { return b == 0 ? a : gcd(b, a % b); }

int main() {
  cout << isTriple(5, 12, 13) << " " << isTriple(4, 5, 6) << endl;      // 1 0
  int s = 8 * 8 + 15 * 15, c = isqrt(s);
  cout << c << " " << (c * c == s) << endl;                              // 17 1: the hypotenuse is whole

  for (int h = 1; h <= 30; h++)                                          // primitive triples up to 30
    for (int a = 1; a < h; a++)
      for (int b = a + 1; b < h; b++)
        if (isTriple(a, b, h) && gcd(a, b) == 1) cout << "(" << a << "," << b << "," << h << ") ";
  cout << endl;

  int count = 0;                                                         // Euclid's formula up to 100
  for (int m = 2; m * m + 1 <= 100; m++)
    for (int n = 1; n < m; n++)
      if ((m - n) % 2 == 1 && gcd(m, n) == 1 && m * m + n * n <= 100) count++;
  cout << count << endl;

  printf("%.1f %.1f\n", sqrt(3.0 * 3 + 4 * 4), sqrt(2.0 * 2 + 3 * 3 + 6 * 6));   // 2D and 3D lengths
  cout << (sqrt(1.2 * 1.2 + 3.5 * 3.5) == 3.7) << endl;                  // 0: the float sum rounds
  double big = 1e200;
  cout << (big * big > 1e308) << endl;                                   // 1: squaring overflows to inf
  return 0;
}

/* output:
1 0
17 1
(3,4,5) (5,12,13) (8,15,17) (7,24,25) (20,21,29)
16
5.0 7.0
0
1
*/

The pitfalls, in order of how often they bite:

Who discovered the Pythagorean theorem?

No single person discovered it: the relation was used in Babylonia, India and China long before Pythagoras, and the first proofs we can read come from Euclid.

• c. 1800 BCE. The Babylonian clay tablet Plimpton 322 lists what appear to be Pythagorean triples, such as (119,120,169), so the relation was in use a thousand years before Greek geometry.
• c. 800–500 BCE. The Indian Sulba Sutras, rules for building altars, state that the rope along the diagonal of a rectangle produces the area of the squares on both sides together.
• c. 570–495 BCE. Pythagoras of Samos, in southern Italy, founded the school that bears his name. Nothing he wrote survives, and the credit for the theorem comes from writers centuries later.
• Before the 1st century BCE. The Chinese Zhoubi Suanjing explains the 3–4–5 triangle and a diagram with a tilted square, the gougu (leg-and-leg) theorem.
• c. 300 BCE. Euclid's Elements proves the theorem as I.47 and its converse as I.48.
• 12th century. Bhaskara II draws the rearrangement diagram with the single word "Behold!".
• 1876. Garfield's trapezoid proof appears in print.
• 1995. Wiles proves Fermat's Last Theorem, the answer to the question of whether the Pythagorean equation works with exponent three or more.

A popular story says that the Greeks were shaken by the discovery that the diagonal of a square is not a ratio of whole numbers, the first irrational number, which the theorem makes unavoidable: see the proof that the square root of 2 is irrational.

What mistakes do people make with the Pythagorean theorem?

The most common mistakes with the theorem are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: the Pythagorean theorem at a glance

• Statement: right triangle with legs a,b and hypotenuse c: a^2+b^2=c^2; c=√(a^2+b^2), a leg is √(c^2-a^2).
• Proofs: rearranging four triangles in an (a+b) square; similar triangles (a^2=cp, b^2=cq); Euclid's shearing; Garfield's trapezoid.
• Converse: a^2+b^2=c^2⇒ right angle; the 3–4–5 rope, the diagonal check.
• Triples: (m^2-n^2,2mn,m^2+n^2); primitive when \gcd(m,n)=1 and opposite parity; 60| abc; Berggren tree from (3,4,5).
• Distance: √(\Delta x^2+\Delta y^2); box diagonal √(l^2+w^2+h^2).
• Generalizations: law of cosines, similar shapes on the sides, any dimension, spheres (\cos c=\cos a\cos b), Fermat's Last Theorem.
• Code: math.hypot, whole-number tests with isqrt, never float equality.

Glossary: key terms

Pythagorean theorem

The rule that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs.

Legs of a right triangle

The two sides of a right triangle that meet at the right angle, as opposed to the hypotenuse.

Converse of the Pythagorean theorem

The statement that a triangle whose sides satisfy a squared plus b squared equals c squared has a right angle opposite the side c.

Pythagorean triple

Three positive whole numbers a, b and c with a squared plus b squared equal to c squared, such as 3, 4 and 5.

Primitive Pythagorean triple

A Pythagorean triple whose three numbers have no common factor greater than 1, so it is not a multiple of a smaller triple.

Euclid's formula

The rule that for whole numbers m greater than n the numbers m squared minus n squared, 2mn and m squared plus n squared form a Pythagorean triple.

Berggren tree

The tree that starts at the triple 3, 4, 5 and gives every primitive Pythagorean triple exactly once through three rules that produce three children.

Distance formula

The formula that gives the distance between two points as the square root of the sum of the squared coordinate differences, found with the Pythagorean theorem.

Fermat's Last Theorem

The theorem that x to the power n plus y to the power n equals z to the power n has no solution in positive whole numbers when n is greater than 2.

Frequently asked questions

References

1. Thomas L. Heath (translator) (1908). The Thirteen Books of Euclid's Elements (2nd ed.), Book I, propositions 47 and 48. Cambridge University Press.
2. Eli Maor (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton University Press.
3. Elisha Scott Loomis (1940). The Pythagorean Proposition. National Council of Teachers of Mathematics.
4. Otto Neugebauer and Abraham Sachs (1945). Mathematical Cuneiform Texts (the tablet Plimpton 322). American Oriental Society.
5. James A. Garfield (1876). A new proof of the Pythagorean theorem. New-England Journal of Education, 3(14), 161.
6. Python Software Foundation. The Python Standard Library: math (hypot, dist and isqrt). docs.python.org. https://docs.python.org/3/library/math.html.

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• Quadrilaterals: Types, Properties, Perimeter and Area: Quadrilaterals explained: square, rectangle, rhombus, parallelogram, kite and trapezoid, their properties, perimeter, area and coordinates, with practice.
• Irrational Numbers: Definition, Examples, Proofs and Approximations: Irrational numbers explained: why √2 is irrational, which roots are, sums and products, π, e, transcendental numbers and fraction approximations, with practice.