Rightangled triangles with whole number sides have fascinated both professional mathematicians
and numberenthusiasts since well before 300 BC when Pythagoras wrote about his famous "theorem".
The oldest mathematical document in the world, a little slab of clay that would fit in your hand,
is a list of such triangles. So what is so fascinating about them? This page starts from scratch
and has lots of facts and figures with several online calculators to help in your own
investigations.
What's on this page
 mens the link next to it is to an interactive Calculator on this page
Rightangled Triangles and Pythagoras' Theorem
Pythagoras and Pythagoras' Theorem
Pythagoras was a mathematician born in Greece in about 570 BC. He was interested in
mathematics, science and philosophy. He is known to most people because of
the Pythagoras Theorem that is about a property of all
triangles with a rightangle (an angle of 90°):
If a triangle has one angle which is a rightangle (i.e. 90°)
then there is a special relationship between the lengths of its three sides:
If the longest side (called the hypotenuse) is
h
and the other two sides (next to the
right angle) are called
a and
b, then:
a^{2} + b^{2} = h^{2}
Pythagoras' Theorem
or,
the square of the longest side is the same as the sum of the squares of
the other two sides.
a^{2} + b^{2} = h^{2}
is only true for rightangled triangles.
 If all the angles of a triangle are less than 90° then the square of the
longest side is less than the sum of the squares of the other two sides;
 If one the angles of a triangle is greater than 90° then the square of the
longest side is greater than the sum of the squares of the other two sides.
For example, if the two shorter sides of a rightangled triangle are 2 cm and 3 cm, what is
the length of the longest side?
If the longest side is h, then, by Pythagoras' Theorem, we have:

h^{2}  =  2^{2} + 3^{2} = 13 
h  =  13 = 3·60555 
The 345 Triangle
In the example above, we chose two wholenumber sides and found the longest
side, which was not a whole number.
It is perhaps surprising that there are some rightangled triangles where
all three sides are whole numbers called Pythagorean Triangles.
The three whole number sidelengths are called
a Pythagorean triple or triad.
An example is a = 3, b = 4 and
h = 5, called "the 345 triangle". We can check it as follows:
3^{2}+4^{2} = 9 + 16 = 25 = 5^{2} so
a^{2} + b^{2} = h^{2}.
This triple was known to the Babylonians (who lived in the area of presentday Iraq and Iran)
even as long as 5000 years ago! Perhaps they used it to make a rightangled triangle so they could
make true rightangles when constructing buildings  we do not know for certain.
It is very easy to use this to get a rightangle using equallyspaced knots
in a piece of rope, with the help of two friends.
If you hold the ends of the rope together together and one friend holds the 4th knot and
and the other the 7th knot and you all then tug to stretch the rope into a triangle,
you will have the 345 triangle
that has a true rightangle in it.
The rope can be as long as you like
so you could lay out an accurate rightangle of any size.
Test a Triangle  is it Pythagorean?
Here is a little calculator that, given any two sides of a rightangled triangle
will compute the third. Or you can give it all three sides to check.
It will check that it is rightangled and, if so, if it is Pythagorean
(all the sides are integers).
Here a and b are the
two legs  the sides surrounding the rightangle and h is the longest side,
the hypotenuse:
More Pythagorean Triples
Is the 345 the only Pythagorean Triple?
No, because we can double the length of the sides of the 345 triangle
to and still have a rightangled triangle e.g
Its sides will be 6810 and we can check that
10^{2} = 6^{2} + 8^{2}.
Continuing this process by tripling 345 and quadrupling
and so on we have an infinite number of Pythagorean triples:
3  4  5 
6  8  10 
12  16  20 
15  20  25 
18  24  30 
or we can take the serires of multiples 1, 11, 111, 1111, etc and get a pattern:
3  4  5 
33  44  55 
333  444  555 
3333  4444  5555 
 ... 
All of these will have the same shape (have the same angles)
but differ in size.
Are there any differentlyshaped rightangled triangles with whole
number sides?
Yes; one is 5, 12, 13 and another is 7, 24, 25.
How do we find these? Is there a systematic method?
In the next section we answer the question
Is there is a formula for generating Pythagorean triples?
The mn Formula for Pythagorean Triples
Yes  we can generate Pythagorean Triples by supplying two different positive integer
values for m and n in this diagram.
You can multiply out these terms and check that
( m^{2} – n^{2} )^{2} + (2 m n)^{2} = ( m^{2} + n^{2} )^{2}
Once we have found one triple, we have seen that we can generate many others by just
scaling up all the sides by the same factor.
A Pythagorean triple which is not a multiple of a smaller one is called a primitive Pythagorean
triple.
Are all the Pythagorean triples of this mn form?
The bad news is that the answer is "No", but the good news is
that all primitive Pythagorean triples are generated by some m and n values
in the formula above!
The formula using m and n will not give
all triples since it misses some of the nonprimitive ones, such as 91215. This is a
Pythagorean triple since, as a triangle, is it just 3 times the 345 triangle (by which we mean
that we just triple the lengths of each side of a 345 triangle, which we already know is
rightangled).
But 9, 12, 15 is missed by our m,n formula because:
Our formula said m and n were positive whole numbers and the Pythagorean triple was
m^{2}  n^{2} , 2mn , m^{2} + n^{2}
and, since we want (positive) whole number values in our triple, then m>n
(otherwise the first number in the triple is negative).
The 2mn value is one of the sides and the only even side in 91215 is 12,
so 12 = 2mn.
Hence mn = 6 and m>n, so we can only have two cases:
 m = 6 with n = 1 OR
 m = 3 and so n = 2
The first case gives the triple 35, 12, 37 and the second case gives 5, 12, 13, neither of which is
the 9, 12, 15 triple.
A mn Pythagorean Triple generator
Here is a calculator to compute the sides of a triangle using the formula above  just type
in the values for m and n. Remember that the formula will find all the Primitive Triples but it will NOT
find all the multiple ones. The calculator will tell you if your values of m and n generate a Primitive
triple or a multiple of a primitive triple.
Using just the primitive triples generated by the formula, the Calculator will find all triples with a
longest side (hypotenuse) of a particular value or within a range of values.
The mn formula and a proof are given as Theorem 225 in
The Theory of Numbers by G Hardy and E M Wright,
(5th edition, 1980) OUP, 442 pages. This is a classic book that is well worth studying but it
does tend to be at university undergraduate level some of the time.
Pythagorean Triples
Now we have a generator for all primitive triples, we can use it to
compute Pythagorean Triples in lots of different ways. This calculator
generates primitive triangles from m and n values (and the multiples of each
if required) so will be slower for larger values of the hypotenuse.
We can use the Calculator here to find all Pythagorean triangles with longest side up to 100, together with
the length of the perimeter P (the sum of all three sides) and its area A
which, since they are all
rightangled triangles, is just half the product of the two sides (those that form the rightangle):
All Pythagorean Triples with sides up to 100
arranged in order of hypotenuse (longest side):
with
Perimeter (a+b+h) and
Area (ab)/2
Triad  Primitive?  P  A 
3, 4, 5  primitive  12  6 
6, 8, 10  2x 3, 4, 5  24  24 
5, 12, 13  primitive  30  30 
9, 12, 15  3x 3, 4, 5  36  54 
8, 15, 17  primitive  40  60 
12, 16, 20  4x 3, 4, 5  48  96 
15, 20, 25  5x 3, 4, 5  60  150 
7, 24, 25  primitive  56  84 
10, 24, 26  2x 5, 12, 13  60  120 
20, 21, 29  primitive  70  210 
18, 24, 30  6x 3, 4, 5  72  216 
16, 30, 34  2x 8, 15, 17  80  240 
21, 28, 35  7x 3, 4, 5  84  294 
12, 35, 37  primitive  84  210 
15, 36, 39  3x 5, 12, 13  90  270 
24, 32, 40  8x 3, 4, 5  96  384 
9, 40, 41  primitive  90  180 
27, 36, 45  9x 3, 4, 5  108  486 
30, 40, 50  10x 3, 4, 5  120  600 
14, 48, 50  2x 7, 24, 25  112  336 
24, 45, 51  3x 8, 15, 17  120  540 
20, 48, 52  4x 5, 12, 13  120  480 
28, 45, 53  primitive  126  630 
33, 44, 55  11x 3, 4, 5  132  726 
40, 42, 58  2x 20, 21, 29  140  840 
36, 48, 60  12x 3, 4, 5  144  864 

Triad  Primitive?  P  A 
11, 60, 61  primitive  132  330 
39, 52, 65  13x 3, 4, 5  156  1014 
25, 60, 65  5x 5, 12, 13  150  750 
33, 56, 65  primitive  154  924 
16, 63, 65  primitive  144  504 
32, 60, 68  4x 8, 15, 17  160  960 
42, 56, 70  14x 3, 4, 5  168  1176 
48, 55, 73  primitive  176  1320 
24, 70, 74  2x 12, 35, 37  168  840 
45, 60, 75  15x 3, 4, 5  180  1350 
21, 72, 75  3x 7, 24, 25  168  756 
30, 72, 78  6x 5, 12, 13  180  1080 
48, 64, 80  16x 3, 4, 5  192  1536 
18, 80, 82  2x 9, 40, 41  180  720 
51, 68, 85  17x 3, 4, 5  204  1734 
40, 75, 85  5x 8, 15, 17  200  1500 
36, 77, 85  primitive  198  1386 
13, 84, 85  primitive  182  546 
60, 63, 87  3x 20, 21, 29  210  1890 
39, 80, 89  primitive  208  1560 
54, 72, 90  18x 3, 4, 5  216  1944 
35, 84, 91  7x 5, 12, 13  210  1470 
57, 76, 95  19x 3, 4, 5  228  2166 
65, 72, 97  primitive  234  2340 
60, 80, 100  20x 3, 4, 5  240  2400 
28, 96, 100  4x 7, 24, 25  224  1344 

See also Douglas Butler's web page
with a list of 2098 triples
with hypotenuse up to 2100.
Patterns in Pythagorean Triples
Let's call the two sides of the triangle that form the rightangle, its legs and use the letters
a and b. The hypotenuse is the longest side opposite the rightangle and we will often use
h for it.
The two legs and the hypotenuse are the three sides of the triangle,
triple or triad a,b,h.
Different authors use different ways of writing triads such as abh but we will use a,b,h on this page.
The series of lengths of the hypotenuse of primitive Pythagorean triangles begins 5, 13, 17, 25, 29, 37, 41 and is
A020882 in Sloane's
Online Encyclopedia of Integer Sequences. It will
contain 65 twice  the smallest number that can be the hypotenuse of more than one primitive Pythagorean triangle.
The series of numbers that are the hypotenuse of more than one primitive Pythagorean triangle is
65, 85, 145, 185, 205, 221, 265, 305,... A024409
There are lots of patterns in the list of Pythagorean Triples above. To start off your investigations here are
a few.
Hypotenuse and Longest side are consecutive
The first and simplest Pythagorean triangle is the 3, 4, 5
triangle. Also near the top of the list is 5, 12, 13.
In both of these the longest side and the hypotenuse are consecutive integers.
Are there any more like this?
The list tells us there are and we have the following:
3,  4,  5 
5,  12,  13 
7,  24,  25 
9,  40,  41 
11,  60,  61 
Can you spot the pattern?
It is not immediately obvious, but you will have noticed that the smallest sides
are the odd numbers 3, 5, 7, 9,... So the smallest sides are of the form 2i+1.
The other sides, as a series are 4, 12, 24, 40, 60... Can we find a formula here?
We notice they are all multiples of 4: 4x1, 4x3, 4x6, 4x10, 4x15,.. . The series of multiples:
1, 3, 6, 10, 15,... is the Triangle Numbers (look them up in Neil Sloane's
Online Encyclopedia of Integer Sequences)
with a formula i(i+1)/2.
So our second sides are 4 times each of these, or, simply, just 2i(i+1).
The third side is just one more than the second side: 2i(i+1)+1,
so our formula is as follows:
Hypotenuse is one more than the longest side:
shortest side = 2i+1;
longest side = 2i(i+1);
hypotenuse = 1+2i(i+1)
Check now that the sum of the squares of the two sides is the same as the square of the hypotenuse (Pythagoras's Theorem).
i  a:2i+1  b:2i(i+1)  h=b+1 
1  3  2x1x2=4  5 
2  5  2x2x3=12  13 
3  7  2x3x4=24  25 
4  9  2x4x5=40  41 
5  11  2x5x6=60  61 
6  13  2x6x7=84  85 
7  15  2x7x8=112  113 
Alternatively, let's look at the mn values for each of these triples. Since the
hypotenuse is one more than a leg, the 3 sides have no common factor so are primitive and therefore
all of them do have mn values:
Triple  m  n 
3, 4, 5  2  1 
5, 12, 13  3  2 
7, 24, 25  4  3 
9, 40, 41  5  4 
11, 60, 61  6  5 
It is easy to see that m = n+1.
The mn formula in this case gives
a = m^{2} – n^{2} = (n+1)^{2} – n^{2} = 2 n + 1
b = 2 m n = 2 (n+1) n = 2 n^{2} + 2 n
h = m^{2} + n^{2} = (n+1)^{2} + n^{2} = 2 n^{2} + 2 n + 1
So h is b+1 and the pattern is always true:
if m = n+1 in the mn formula
then it generates a (primitive) triple with hypotenuse = 1 + the longest leg.
Are these all there are? Perhaps there are other mn values with a leg and hypotenuse consecutive numbers.
In fact, they are all given by the formula above because:
The triangles must be primitive so we know they have an mn form.
h = m^{2} + n^{2} could be either one more than
either m^{2} – n^{2}
or 2 m n:
 If h = m^{2} + n^{2} = (m^{2} – n^{2}) + 1
then, taking m^{2} from both sides:
n^{2} = – n^{2} + 1
2 n^{2} = 1 or n^{2} = ^{1}/_{2}
and this is not possible for a whole number n
So h is never a + 1.
 If h = m^{2} + n^{2} = 2 m n + 1 then
m^{2} – 2 m n + n^{2} = 1 which is the same as
(m – n)^{2} = 1. Therefore
m – n = 1 or m – n = –1
But the hypotenuse is a positive number and is m^{2} – n^{2}
so we must have so m>n and so m – n cannot be –1
The only condition we can have is that m – n = 1 or m = n + 1
 there are no other triangles with hypotenuse one more than a leg except those generated by consecutive
m n values in the mn formula.
More information on these series:
 Shortest legs: 3, 5, 7, 9, 11, 13,... A081874
 Longest legs: 4, 12, 24, 40, 60, 84,... A046092
 Hypotenuses: 5, 13, 25, 41, 61,... A001844
An easy method of writing down a series of Triples
Look at the following series of Pythagorean triples. It is easy to spot the pattern and to
remember it. If you then write it down to show your friends it will
look as if you have impressive calculating skills!
21  220  221 
201  20200  20201 
2001  2002000  2002001 
20001  200020000  200020001 
It uses n+1 for m
in the formula and then lets n be powers of 10. This simplifies the triples
to be 2n+1, 2n(n+1), 2n^{2}+2n+1.
Can you find another simple method like this that always produces Pythagorean Triangles (not necessarily with consecutive sides)?

Solution to Problem: Find a Scheme for writing mechanically an unlimited number of Pythagorean Triangles
M Willey, E C Kennedy, American Mathematical Monthly vol 41 (1934) page 330.

There is a Calculator below that you can use to generate
many more simple patterns like this one.
The two legs are consecutive
Also in the 3, 4, 5 triple, the two legs of the triangle a and
b are
consecutive, b = a+1. Are there any more like this?
Yes! 20, 21, 29.
Although the list above does not contain any more, there are larger examples:
3, 4, 5 20, 21, 29 119, 120, 169 696, 697, 985
Because the two legs are consecutive numbers, they will have no common factor so all of these will be
primitive. We can therefore find certain special values for m and n
in the mn formula above. Here is the same list with their mn values:
m  n  a=m^{2}n^{2} 
b=a+1=2mn 
h=m^{2}+n^{2} 
2  1  3  4  5 
5  2  21  20  29 
12  5  119  120  169 
29  12  697  696  985 
This already suggests a way that we can use the generators
m and n
for one triple to find generators for the next.
See if you can find how and also find a formula for this pattern of triples.
Kayne Johnston aged 13 has also found the neat pattern to compute this table without using
the m and n generators, each row being computed simply from the two rows before:
the next row in the table has a smallest side
that is 6 times the previous smallest side minus
the smallest side before that plus 2.
For instance, after the first two rows:
a  b=a+1 
h=(a^{2}+b^{2}) 
3  4  5 
21  20  29 
the left hand column (the smallest side) is six times 21
minus the one before (3) plus 2:
6x203+2 = 1203+2 = 119.
The other side is just one more than the smallest, so here is it 120.
Can you find a similar method to compute the hypotenuse but without using Pythagoras Theorem on the
two sides?
The series here are:
 The shortest sides are 3, 20, 119, 696,... A001652
 The second legs are 4, 21, 120, 697,... A046090
 The hypotenuses are 5, 29, 169, 985,... A001653
Beiler (see references at the foot of this page) gives a formula for these triples
so that the r^{th} triple in
this list is given directly in terms of r.
More patterns
There are more patterns in the Pythagorean Triples Table above.
For instance
there are primitive triangles whose longest side and hypotenuse
differ by 2, such as 8, 15, 17 and 12, 35, 37
and many more.
What is the mathematical pattern in these triples?
Another idea is to take the formula and find special cases,
remembering that the formula does not generate all
Pythagorean triples.
For instance, let n=1. We then have triples
m^{2}–1, 2m, m^{2}+1, although we have
to make the restriction that m>1 for the hypotenuse to be a positive number:
m  m^{2}–1  2m  m^{2}+1 
2  3  4  5 
3  8  6  10 
4  17  8  19 
5  24  10  26 
6  35  12  37 
Notice that not all of these are primitive and also that there are other triples with a side two less than the hypotenuse that
are not in this list.
Can any number be a side in some Pythagorean Triangle?
Note that here we use the terms leg, side, hypotenuse as follows:
there are two legs and a hypotenuse making the 3 sides of each
Pythagorean triangle.
The Number of Pythagorean Triangles having a side n
These sequences are the counts for n=1, 2, 3,... in each of the 6 categories:
so that 0, 0, 1, 1, 2,.. means
0 triangles for n=1, 0 for n=2, 1 for n=3, 1 for n=4, 2 for n=5 etc
 Primitive  All 
as a leg  A024361
0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1,.. 
A046079
0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3,... 
as a hypotenuse 
A024362
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,... 
A046080
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0,...

total 
A024363
0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 2, 1,... 
A046081
0, 0, 1, 1, 2, 1, 1, 2, 2, 2, 1, 4, 2, 1, 5, 3,... 
It looks like every fourth number after 2, namely 6, 10, 14, 28,... cannot be the side of a primitive triangle.
Also you might guess that there are no gaps in the list of numbers that can be a side of at least one triple.
These are indeed true but not proved here.
From the table above, we can make an ordered list of the numbers themselves that can appear as sides in each category.
The Possible Sides of Pythagorean Triangles
Here the actual sides are listed.
If there is more than one possible Pythagorean triangle with a given side, the side is repeated in these sequences.
The sequence of possible side lengths without repetitions is given in brackets.
 Primitive  All 
Legs 
3, 4, 5, 7, 8, 9, 11, 12, 12, 13, 15, 15, 16,...
A024355
(A042965) 
3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14,...
A009041
(every integer>2) 
Hypotenuses 
5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 65,...
A020882
(A008846) 
5, 10, 13, 15, 17, 20, 25, 25, 26,...
A009000
(A009003) 
Sides 
3, 4, 5, 5, 7, 8, 9, 11, 12, 12, 13, 13,...
A024357
(A042965)

3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12, 12, 13, 13,...
A009070
(every integer>2) 
Number Series in Pythagorean Triangles
Other series of interest here are:
 A020883 Longest legs in primitive triples:
4, 12, 15, 21, 24, 35, 40, 45, 55, 56,... (or
A024354 without duplicates or
A024360
as a count of the number of triangles with longest side n or
A0244410
longest legs in more than one primitive triangle)
 A020884 Shortest legs in primitive triples:
3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 20,...
(A024352 without repetitions
or A024359
as a list of counts of the number of triangles with each shortest side n or
A0244411
shortest legs in more than one primitive triangle)
Except for 0 and 1, these are all the numbers that are the difference of two squares or,
equivalently, every number that when divided by 4 has a remainder of 0, 1 or 3
A042965).
 A024409
Hypotenuse of more than one primitive triangle
 A024406 Area of primitive triangles:
6, 30, 60, 84, 180, 210, 210, 330,... (
A020885 is areas divided by 6,
A024407
areas of mroe than one primitive triangle
since all such areas are a multiple of 6)
 A024364 The perimeters of primitive triangles:
12, 30, 40, 56, 70, 84, 90,... (A024408
primeters of more than one primitibe triangle); also
A099829 Smallest perimeter of n Pythagorean triangles
 A006593 the smallest number that
occurs as a side in n triads: 3, 5, 16, 12, 15, 125, 24,... (so 3 is the smallest side in just 1 triad,
5 is the smallest that occurs in exactly 2 triads, 16 in 3 triads, etc)
Pythagorean Triangles with Equal Perimeters

If I have a piece of string of length n, how many rightangled triangles can I make from it if the sides have integer lengths?
The diagram here shows the only configuration if the rope has length 3+4+5=12. The length of the rope is the perimeter of the
triangle.
A009096 lists the perimeters of all Pythagorean
triangles in order: 12, 24, 30, 36, 40, 48, 56, 60, 60, 70,...
(A010814 without repetitions)
 The same problem but this time we want the smallest length of string that makes exactly n different rightangled triangles.
It is 12 for a unique triangle: 3, 4, 5 whereas a length of 60 can make 2 different triangles:
5x 3, 4, 5 and 2x 5, 12, 13
but it has to be 120 for 3 triangles: 10x 3, 4, 5, 4x 5, 12, 13,
24, 45, 5, so the series is 12, 60, 120,...
A099830
 A099831 Perimeters of exactly 2 Pythagorean triangles: the
length of the rope if we can form exactly 2 different Pythagorean triangles using it. We saw in the previous problem that the smallest is 60,
so what are the other values? They are
60, 84, 90, 132, 144, 210,..
since 84=7x 3, 4, 5 and 12, 35, 37, the next is 90 and so on.
 A099832 Perimeters of exactly 3 Pythagorean triangles: 120 is the smallest
and also,in order, we have 120, 168, 180, 252, 280, ...
 A099833 Perimeters of exactly 4 Pythagorean triangles:
240, 360, 480, 504, 630,...
The Incircle and Inradius
We can draw a circle touching all 3 sides of any triangle, called the incircle
with radius the inradius usually denoted by r and centre the incentre.
From the symmetry of the circle, a line from its centre to each vertex of the triangle will halve each of the
angles in the triangle.
Lines from the incentre to the vertices (shown in
blue here) divide the triangle into three smaller ones, each having the same
height, r on a base of one side of the whole triangle.
The area of a triangle is one half of the base of the triangle times its height.
So the three separate areas sum to the whole area:
area =  a r  +  b r  +  c r  = r  a + b + c 
   
2  2  2  2 
The sum of the sides of a triangle is the length of its perimeter.
We now have a simple formula for the inradius, r of any triangle:
r =  2 area 

perimeter 
In a rightangled triangle, the area is just half the product of the two legs,
a b / 2 or
2 area = a b so the formula for r is even simpler:
r =  a b   a + b + h 
 in a rightangled triangle 
Since all primitive rightangled Pythagorean triangles can be derived using the mn
formula above, then, in terms of m and n we have
r =  (m^{2}–n^{2}) 2mn  =  (m^{2}–n^{2}) 2mn  =  (m–n)(m+n)2mn  = (m–n)n 
  
m^{2}–n^{2} + 2mn + m^{2}+n^{2}  2m^{2} + 2mn  2m(m+n) 
In primitive rightangled triangles, the inradius, r
is therefore always an integer because m and n are.
Nonprimitive triangles are just multiples of primitives, so their inradius is an integer too.
We now have the general rule that:
the inradius is always a whole number in all Pythagorean triangles!
A more general Pythagorean Triple Calculator
This calculator will find Pythagorean triples for you, either primitive or all with
any combination of sides with a fixed value or in a given range of values.
You can find the actual triples or else just count
the number found.
If you give a range of values, the total in that range can be counted (Total count of)
or else
a separate count with one count for each value in the range is given if you select
Separately count. Sizes gives a list of those values in the range for which
the requested type of triangle (all or primitive) exist. Values are
repeated where there are different triples of the same size.
For example:
 The total count of triples with a hypotenuse in the range 2030 is 6
 In a List they are:
1: 12, 16, 20 =4x[3, 4, 5] P=48 A=96 r=4 m=4 n=2
2: 15, 20, 25 =5x[3, 4, 5] P=60 A=150 r=5 m=. n=.
3: 7, 24, 25 primitive P=56 A=84 r=3 m=4 n=3
4: 10, 24, 26 =2x[5, 12, 13] P=60 A=120 r=4 m=5 n=1
5: 20, 21, 29 primitive P=70 A=210 r=6 m=5 n=2
6: 18, 24, 30 =6x[3, 4, 5] P=72 A=216 r=6 m=. n=.
 Counted separately, there is one of size 20, 2 of size 25, and one each of 26, 29 and 30
so the separate counts are reported as
1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1: a count for each of the values from 20 to 30.
 The sizes of the 6 triples are 20, 25, 25, 26, 29, 30.
(*) There are very fast algorithms for counting the number of
triangles (primitive or all) given the size of a leg, or a hypotenuse or a side, marked (*) in the list in the Calculator here.
Otherwise, the
triangles are generated and then counted, a process that takes some time for large sides.
Sizes reports the sizes (of the side/perimeter/area requested) in the
given range, so that if a side/perimeter/area is found in
more than one triple, it is reported once for each separate triple.
List lists all the triples found but if you want just one example
use List one of.
The results
are printed in the Results box, triples being given with their area,
perimeter and inradius. This data can easily be selected and copied to be used
in other applications or as text.
Here is the triple generator which can search for various conditions on its sides and has some
very fast counting algorithms for some cases:
Finding a Pythagorean Triangle approximating a given Angle
Can we find a Pythagorean triangle with a given angle? Sometimes this may not be possible with small numbers
but we will always be able to find some Pythagorean triangles with an angle almost equal to
any angle you require.
Here is a Calculator to find a primitive triangle with better and better approximations to a given angle.
It only generates primitive triangles since all its multiples have identical angles but bigger sides.
You can use Pi in the input box e.g. for the angle Pi/3 (radians).
If you want Pythagorean triangles
with a specific ratio of sides, e.g. 1/3, then use the angle with that sine or cosine or tangent:
e.g.
 asin(1/3) is the angle (radians) whose sine is 1/3, i.e. the ratio of the leg opposite the angle to
the hypotenuse;
 acos( (sqrt(5)1)/2 ) for the ratio of the leg next to the angle to the hypotenuse;
 atan(3/4) for the ratio of the leg opposite to the leg next to the angle.
Calculations are slightly more accurate if radian measure is used.
Further Triple Patterns
The AngleinTriple Calculator above finds some interesting number patterns too.
For instance, there is
no Pythagorean triangle with an angle of 45° (type in 45 into the "angle of..." box,
make sure degrees is on and then click on the Find button)
and if we ask the Calculator to find approximations
it finds the sequence with legs differing by one that we
found above.
If we try it on a series of angles such as 0.1, 0.01 and 0.001 radians,
we discover two series
of easilyremembered and visually striking patterns
as in An easy method of writing down a
series of triples above:
399  40  401   180  19  181 
39999  400  40001   19800  199  19801 
3999999  4000  4000001   1998000  1999  1998001 
399999999  40000  400000001   199980000  19999  199980001 
Try it with 0.2, 0.02, 0.002 radians and you'll find at least two more
patterns like this!
40  9  41   99  20  101 
4900  99  4901  9999  200  10001 
499000  999  499001  999999  2000  1000001 
49990000  9999  49990001  99999999  20000  100000001 
There are some more complicated patterns here too:
88501  17940  90301      
899850001  17999400  900030001  446930400  8939801  447019801 
8999985000001  17999994000  9000003000001  4496993004000  8993998001  4497001998001 
With 0.3, 0, 03 and 0,.003 there are patterns in the first four approximations:
12  5  13   24  7  25 
2112  65  2113  2244  67  2245 
221112  665  221113  222444  667  222445 
22211112  6665  22211113  22224444  6667  22224445 
532  165  557   391  120  409 
55432  1665  55457  39991  1200  40009 
5554432  16665  5554457  3999991  12000  4000009 
555544432  166665  555544457  399999991  120000  400000009 
The first triple in the final series is the one suggested by the rest of the pattern. It is indeed
a Pythagorean triple  check it with the Test a Triangle  is it Pythagorean?
calculator above.
The series of angles 0.4, 0.04, 0.004 and 0.0004 radians generates:
12  5  13   24  10  26 
1200  49  1201  2499  100  2501 
124500  499  124501  249999  1000  250001 
12495000  4999  12495001  24999999  10000  25000001 
The 0.5 radians sequence gives:
     15  8  17 
760  39  761  1599  80  1601 
79600  399  79601  159999  800  160001 
7996000  3999  7996001  15999999  8000  16000001 
and with 0.6 radiansand its tenths:
4  3  5   91  60  109 
544  33  545  9991  600  10009 
55444  333  55445  999991  6000  1000009 
5554444  3333  5554445  99999991  60000  100000009 
380  261  461 
5436800  326601  5446601 
55443668000  332666001  55444666001 
555444366680000  333266660001  555444466660001 
0.7 radians seems to give only one simple pattern:
351  280  449 
39951  2800  40049 
3999951  28000  4000049 
399999951  280000  400000049 
but 0.8 gives two:
     624  50  626 
31000  249  31001  62499  500  62501 
3122500  2499  3122501  6249999  5000  6250001 
312475000  24999  312475001  624999999  50000  625000001 
and 0.9 gives several:
4  3  5   4  3  5   3  4  5 
220  21  221  264  23  265  483  44  485 
24420  221  24421  24864  223  24865  49283  444  49285 
2466420  2221  2466421  2470864  2223  2470865  4937283  4444  4937285 
246886420  22221  246886421  246930864  22223  246930865  493817283  44444  493817285 
20  21  29   48  55  73   319  360  481 
2240  201  2249  6148  555  6173  39919  3600  40081 
222440  2001  222449  617148  5555  617173  3999919  36000  4000081 
22224440  20001  22224449  61727148  55555  61727173  399999919  360000  400000081 
2222244440  200001  2222244449  6172827148  555555  6172827173  39999999919  3600000  40000000081 
0.010 will again give the first example above for 0.1.
However, do use the Calculator above and repeat the experiment. This time you will probably notice at least
two more pattern series to add to your collection!
How about 0.11, 0.011, 0.0011, ...
and then 0.12, 0.012, 0.0012, ...
and so on?
Also try 2/3, 2/30, 2/300, etc (the Calculator handles expressions as input)
or you can input it as 0.6666, 0.06666, 0.006666, etc
which has a one very simple pattern in particular.
You can also do this with any other (small) sequence of numbers making a decimal.
Remember though that the biggest angle is 90° which is 1.57079632679489 radians.
The reason the patterns are so "obvious" above is that our numbers are written in base 10 and we are taking
angles onetenth as large each time.
An interesting mathematical Project is to find formulae for each of these series.
It will then be easy to verify that
all the triples in the series are Pythagorean by
summing squares of the two legs and checking it equals the square on the hypotenuse.
You might even be able to find a formula that encompasses several of the series above.
What else can you find?
Email me at the address at the foot of this page
and I'll add any interesting triples' series that you find, with your name.
The Shapes and Sizes of Pythagorean Triangles P Shiu, Mathematical Gazette
vol 67 (1983) pages 3338. This describes the algorithm behind the anglefinder calculator above. To find a
Pythagorean triangle with angles close to θ let
u = tan(θ)+ sec(θ) and find its
continued fraction. If the
successive convergents to it are m_{k}/n_{k} then
a suitable Pythagorean triangle is
x = 2 m_{k} n_{k},
y = m_{k}^{2} – n_{k}^{2},
z = m_{k}^{2} + n_{k}^{2}.
More Curious Number Facts about Pythagorean Triangles
In every Pythagorean triangle the following 6 facts are always true:
 one side is a multiple of 3
 one side is a multiple of 4
 one side is a multiple of 5
 the product of the two legs is always a multiple of 12
 the area is always a multiple of 6
 the product of all three sides is always a multiple of 60
We can always find a Pythagorean triangle with a side of any given length (bigger than 2).
Pythagorean Triangles and Egyptian Fractions
Egyptian Fractions (also called Babylonian fractions) is
the method of writing fractional values as
used by the ancient Egyptians who built the Pyramids and before them the ancient Babylonians from whom
we get our time measurements of seconds, minutes and hours and also our 360° in a full turn. They did not use
the ratio of two whole numbers as we do, e.g. 4/5 (the ratio of 4 to 5 is "fourfifths" and is also 4 divided by 5.
Instead they used a sum if unit fractions, so that 3/4 would be 1/2 + 1/4, a sum of two unit different fractions,
that is fractions
that we would write as 1/n but without repeating any fraction in a sum!
Every fraction a/b can be written as a sum of distinct unit fractions, often in several ways
and these are called Egyptian Fractions or Babylonian Fractions.
So the simplest kinds of fractions are those that can be written as a sum of two unit fractions.
The number of ways we can write 2/n as a sum of two different unit fractions gives the series
1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1,... for n=3 onwards since 2/3=1/2+1/6 is the only way for 2/3,
2/8=1/4=1/12+1/6 = 1/20+1/5, two ways, so the first entry (for n=3) is 1 and the entry for n=8 is 2.
Surprisingly this series is just the number of Pythagorean triangles having n as a leg, series
A046079 in the table above!
I have another page at this site to help you explore more about Egyptian Fractions
Pythagorean Triples and Fibonacci Numbers
The Fibonacci Numbers
are a simple series of numbers that appear in lots of places in nature:
1, 2, 3, 5, 8, ... where each number is the sum of the previous two in the series.
Often mathematicians start tis series with 0 and 1 and we get the series:
0, 1, 1, 2, 3, 5, 8, 13, 21, ... .
To make a Pythagorean Triangle, take any 4 consecutive numbers in this series, such as 1, 1, 2, 3:
 Multiply the middle two numbers and double the result, here 1 and 2 multiply to 2 and double to 4
 this is one side of our Pythagorean Triangle
 Multiply the two outer numbers, here 1 and 3 giving 3  the second side of the Pythagorean triangle
 Add the squares of the inner two numbers to get the third side: here 1^{2} + 2^{2} gives 5, the
hypotenuse
So we have found the 3, 4, 5 triangle.
In fact you can start with any two starting numbers and use the
FibonacciRule (add the latest two to get the next) to generate two more and those
4 numbers will always generate a Pythagorean Triangle. Try it!
Is there a Pythagorean Triangle of Fibonacci Numbers?
It is easy to see that no triangle can exist with all 3 sides being different Fibonacci numbers:
If the sides are a<b<c
then c is at least a + b by the Fibonacci rule. However,
in any triangle the two shorter sides must add to more than the longest side or else the sides will not meet
(think of the longest side as the base and the two shorter sides hinged at the ends of the base).
We know of two Pythagorean triangles with 2 Fibonacci numbers as sides:
3 4 5
5 12 13
It is thought that there are no more but this remains an open question.
Pythagorean Triples Containing Fibonacci Numbers: Solutions for
F_{n2 ± Fk2 = K2} by M BicknellJohnson,
Fibonacci Quarterly 17 (1979) pages 112, (Addenda: page 293).
Pythagorean Triples and Pi
While there does not appear to be any simple formula known for the number of Pythagorean
triangles up to a given size (the methods used in the Calculators on this page depend on the factors of a number),
there are a couple of simple approximations discovered by D H Lehmer in 1900 and they both involve
pi ().

The number of primitive Pythagorean triangles with hypotenuse less that N is approximately
^{N}/_{2}

For example, in the Table of Triples with sides up to 100 above, there were
16 primitive triples with a hypotenuse less than 100. Lehmer's approximation gives
100 / (2x3·14159) = 15·91.
 The number of primitive triangles with a perimeter less than N is
approximately ^{N ln(2)}/_{2}
 ln(2) is the natural log of 2 = that power of e which gives 2.
Since e^{0·693147...}=2 then ln(2)= 0·693147... .
For example,
the same Table above shows all 14 primitive triangles with perimeter less than 200; whereas Lehmer's
formula says this is approximately (200 x 0·693147...) / ^{2}
= 14.04.
It seems remarkable that should appear in this context, but it does
have an amazing tendency to appear in many such formulae for approximations.
D H Lehmer, American Journal of Mathematics 1900, vol 22, page 38.
Pythagorean Triples or Babylonian?
A tiny block of clay, about the size of a postcard (5 inches x 3.5 inches or 12cm x 9cm) with 15 rows
of 4 columns of "numbers" is dated to about 1800 BC and so is probably the world's oldest surviving mathematical artifact.
Plimpton 322 is one of 600 such tablets
donated to Columbia University's Rare Book and Manuscript
Library by George Plimpton and was item 322 in his catalogue, hence its name.
(Have a look at their other treasures too.)
Bill Casselman's page on
The Babylonian tablet Plimpton 322 from University of British Columbia has the best image of the tablet and
an excellent explanation of how to read Babylonian numbers and what the tablet contains.
And what does it contain? A list of Pythagorean Triangles arranged in order of triangles which are
approximately 1 degree apart! They are written in their base 60 scale, and involve base 60 "fractions"
and they were probably used in surveying.
Words and Pictures: New Light on Plimpton 322
by Eleanor Robson in American Mathematical Monthly vol 109 (2002), pages 105120 explores three theories as to the
meaning of the numbers on Plimpton 322, one of which is that it is a trigonometric table.
The Exact Sciences in Antiquity
by Otto Neugebauer, Dover, (1969) 240 pages argues that the Plimpton 322 tablet contains Pythagorean triples for
triangles for each degree from 30 to 45 and so detects several simple errors in the tablet's table.
Puzzles and Problems
 Find the only two Pythagorean triangles with an area equal to their perimeter.
 Find three consecutive numbers which can be the hypotenuses of
Pythagorean triangles.
Can you find four consecutive numbers which are hypotenuses?
What about five?
and how about a set of nine?
 Find a few Pythagorean triangles whose smallest side is a square number
e.g.
9=3^{2}, 12, 15, and 25=5^{2}, 312, 313.
Of the primitive ones in your list, what is special about their m
and n values?
 What about Pythagorean triples having a smallest side which is a cube number?
e.g.
27=3^{3}, 36, 45.
What is special about their m
and n values?
 How many Pythagorean triangles have a side of length 48?
Find a number that can be the side of even more Pythagorean triangles. (Hint: there is 3 others less than 100)
 What is the highest number of triples you can find with the same side in each?
 What is the smallest number that is the hypotenuse of more than one triple?
What is the greatest number of triples you can find with the same hypotenuse?
(*) Can you find one that is not a multiple of 5?
 4/3/05 is a date and also a Pythagorean triple. There is another one this year ('05)  when is it?
Assuming that the years are in this century,
how many other days will have a Pythagorean Triple date if the year is the hypotenuse?
If the date is any set of 3 numbers that are Pythagorean triples, how many dates are there in one century?
 How about a special Pythagorean Triple Time in hours:minutes:seconds? How many are there in a
whole day if we use a 24hour clock with hours from 0 to 23?
 In the Easy method of writing down a series of Triples section, we found a formula for the
pattern given there and used it with n = 10, 100, 1000, ....
What pattern do you get with n = 20, 200, 2000, ...?
... and with n = 30, 300, 3000, ...?
 Find your own Pythagorean Triple Pattern not already mentioned in Further Triple Patterns
above.
Here is another way to do this.
Think of a series of numbers that are like those in the lists above, e.g.
from the
399, 40, 401 pattern we might think of hypotenuses that are in the series
901, 90001, 900001, ... . Plug these numbers into the
Triples generator and see if any patterns
emerge. The hypotenuse searches on 901 and 90001 give:
 Triples with hypotenuse=901:
1: 476, 765, 901 =17x[45, 28, 53] P=2142 A=182070 r=170 m=. n=.
2: 424, 795, 901 =53x[15, 8, 17] P=2120 A=168540 r=159 m=. n=.
3: 451, 780, 901 primitive P=2132 A=175890 r=165 m=26 n=15
4: 60, 899, 901 primitive P=1860 A=26970 r=29 m=30 n=1
 Triples with hypotenuse=90001:
1: 600, 89999, 90001 primitive P=180600 A=26999700 r=299 m=300 n=1
and another pattern jumps out:
899, 60, 901
89999, 600, 90001
 Find a formula for one of the patterns in the Further Triple Patterns.
Links and References for Further Reading

a book 

an article, usually in an academic periodical 

a link to a web page 

Recreations in
the Theory of Numbers  The Queen of Mathematics Entertains
by A H Beiler, Dover, 1964,
was the first book that opened my eyes to the wonderful fun and facts about simple numbers.
There is a whole chapter on Pythagorean triangles: The Eternal Triangle. This book has been in print now for
many years and is a real classic, being both readable and full of interesting facts and tables and certainly accessible
to anyone with an interest in "recreational" mathematics and numbers. The subtitle of the book is
The Queen of Mathematics Entertains
which comes from a quote of Karl Frederich Gauss:
Mathematics is the queen of the sciences and arithmetic the queen of mathematics.
Highly recommended!


Mathematical Recreations
(second revised edition) by Maurice Kraitchik, Dover, 1953,
is another very enjoyable book that will appeal to
anyone who likes "playing with numbers". Apart from a chapter on
the classic numerical pastimes and number puzzles, it has others on Magic Squares, Chess board problems,
Permutations, Geometrical recreations and puzzles and a chapter on the Calendar. But I list it here because of
chapter 4 devoted completely to the Pythagorean Triangles called ArithmeticoGeometrical Questions. It has
the details of the algorithms used in the Calculators on this page.
This continues to be one of my favourite recreational mathematics books, of interest to anyone with
just a basic mathematical knowledge and a love of numbers.
It is available secondhand from as little as less than two US dollars at Amazon.com!


Mathematical Recreations and
Essays W W Rouse Ball, H S M Coxeter, Dover (13th edition 1987), paperback, 428 pages.
This is another
of the few great classics on mathematical recreations many of a geometrical nature. There is a fascinating chapter on
people with the most amazing ability to do arithmetic calculations in their heads. For us here, there is only a short section on
Pythagorean triangles, but, if you have found this web page of interest, I am sure that you
will find much to stimulate your own investigations in this book. It rarely uses mathematics
taught beyond age 16. It really is a book packed full of so many interesting and tantalising titbits of mathematics
that it makes you want to get out a pencil and paper and play with the numbers for yourself.


The Book of Numbers
by John Horton Conway and Richard K. Guy, Copernicus Books (1996), 311 pages, hardback.
This is nothing to do with a book of the Old Testament
as they quip in their introduction, but a collection of interesting mathematics
looking at our attempts to get to grips with the idea of number:
number words in many languages and the many different ways to write numbers as well as
numbers in mathematics. Much of it is at schoolmaths level but some of it goes beyond that
(imaginary, transcendental and infinite numbers). However, don't let that put you off as the book is full of diagrams
and pictures and explanations making it all readily accessible. The chapter on Further Fruitfulness of Fractions
shows how Pythagorean triangles were used by the Babylonians of 1500BC, well before the time of Pythagoras around 600BC
as discovered in a little clay tablet called Plimpton Tablet 322 (now in the Columbia University Library).


Number Theory and Its History
Oystein Ore, Dover (1988), 380 pages, paperback
is a great book if you want to look more seriously at the mathematics of primes and factors, congruences (the arithmetic
of remainders on division)  a topic called Number Theory  all in the context of their history by an excellent writer.
There is a section on the Plimpton Tablet 322 (not 332 as he mistakenly labels his picture of it),
a Babylonian list of Pythagorean triples and how it might have been used by the Babylonians.


The Penguin Dictionary of
Curious and Interesting Numbers David Wells, Penguin (Revised edition 1998), will help
answer some of the puzzle questions above but is a curious and interesting book in its own right! Take a number such as
3.14159.. . No doubt you will recognise it but what about 1634
or 364.2422? (Let your mouse rest on the numbers for the answers!)
And how many number facts do you know about
28? This book is full of wonderful facts about your favourite numbers.


When is n a member of a Pythagorean triple? Dominic and Alfred Vella,
Mathematical Gazette Note 87.04, pages 102105, vol 87 (2003). This, with some others on Pythagorean triples
are available in PDF format from Dominic Vella's
mathematics page.


A new algorithm for generating Pythagorean triples
(PDF file) by R. H. Dye and R. W. D. Nickalls in The Mathematical Gazette (1998), volume 82, pages 8691.
The link is to an online PDF file of the article.


Online Encyclopedia of Integer Sequences
is Neil Sloane's excellent resource for both checking series of integers and also finding out more about each one. It is the
worldwide resource for such information and Neil welcomes any additional new series as well as more information on the
individual sequences.


All the primitive triads
for hypotenuse up to 10000. Michael Samos has produced this text file table together with
the perimeter and area of all the triangles (and other information on the angles too) if you want a complete list to print.


Pythagorean Triples Projects
is Eric Rowland's useful page of further ideas for your own investigations
together with some hints and solutions.

© 19962006 Dr Ron Knott
updated 27 April 2006