# Pythagorean Theorem

Updated 2010-06-10 18:18

## Pythagorean Theorem

In a right triangle with sides of length 0 <= a <= b <= c,
c is the length of the hypotenuse (the side opposite the right angle),
while a and b are the lengths of the legs of the triangle.

In such a right triangle, the following hold:

a2 + b2 = c2,     sqrt[a2 + b2] = c,

0 <= c - b <= a      <= b
0 <= c - b <= c - a <= b <= c <= a + b
0 <= b - a <= c - a <= b

If a, b, c are all positive integers, they form what is called a Pythagorean Triple,
as discussed on http://www.mathsisfun.com/numbers/pythagorean-triples.html

and listed below.

Also listed below are the angles A and B in degrees:
A is the angle opposite side a and adjacent to sides b and c.
B is the angle opposite side b and adjacent to sides a and c.

The angles A and B satisfy the following:
A = sin-1(a/c) = cos-1(b/c) = tan-1(a/b).
B = sin-1(b/c) = cos-1(a/c) = tan-1(b/a).
A + B = 90 degrees.
0 degrees <= A <= 45 degrees <= B <= 90 degrees.

The Pythagorean Triples listed below can be used
to build right triangles with the desired angles A and B.

You could try to memorize the chart below,
or you could use its contents as practice exercises.

## Pythagorean Triples

a
b
c
A in degrees
B in degrees
34536.953.1
5121322.667.4
7242516.373.7
8151728.161.9
9404112.777.3
11606110.479.6
12353718.971.1
1384858.881.2
151121137.682.4
16636514.375.7
171441456.783.3
191801816.084.0
20212943.646.4
209910111.478.6
212202215.584.5
232642655.085.0
241431459.580.5
253123134.685.4
273643654.285.8
28455331.958.1
281951978.281.8
294204213.986.1
314804813.786.3
322552577.282.8
33566530.559.5
335445453.586.5
356126133.386.7
36778525.164.9
363233256.483.6
376846853.186.9
39808926.064.0
397607612.987.1

a
b
c
A in degrees
B in degrees
403994015.784.3
418408412.887.2
439249252.787.3
4411712520.669.4
444834855.284.8
48557341.148.9
485755774.885.2
5114014920.070.0
5216517317.572.5
526756774.485.6
567837854.185.9
5717618517.972.1
609110933.456.6
6022122915.274.8
608999013.886.2
65729742.147.9
6828529313.476.6
6926026914.975.1
7530831713.776.3
7635736512.078.0
8418720524.265.8
8443744510.979.1
8513215732.857.2
8741642511.878.2
8810513740.050.0
925255339.980.1
9347648511.178.9
9516819329.560.5
9624726521.268.8

a
b
c
A in degrees
B in degrees
1006216299.180.9
10415318534.255.8
10520823326.863.2
1056086179.880.2
1087257338.581.5
1116806899.380.7
11525227724.565.5
1168378457.982.1
11912016944.845.2
12020924129.960.1
12039140917.172.9
1238368458.481.6
1249579657.482.6
1299209298.082.0
13247549315.574.5
13315620540.449.6
13535237721.069.0
13627330526.563.5
14017122139.350.7
14540843319.670.4
15234537723.866.2
15546849318.371.7
15666768513.276.8

a
b
c
A in degrees
B in degrees
16023128134.755.3
16124028933.956.1
16553255717.272.8
16842545721.668.4
16877579312.277.8
17528833731.358.7
18029934931.059.0
18451354519.770.3
18567269715.474.6
18934038929.160.9
19574877314.675.4
20060964118.271.8
20339644527.162.9
20425332538.951.1
20582885313.976.1
20722430542.747.3
21591293713.376.7
21671374516.973.1
21745650525.464.6
22045950925.664.4
22527235339.650.4
22832539735.154.9
23152056924.066.0
23282585715.774.3
24055160123.566.5
24894597714.775.3
25227537342.547.5
25966070921.468.6

a
b
c
A in degrees
B in degrees
26065170121.868.2
26138046134.555.5
27373678520.469.6
27649356529.260.8
27944052132.457.6
28035144938.651.4
28075980920.269.8
28781686519.470.6
29730442544.345.7
30058966127.063.0
30190094918.571.5
30843553335.354.7
31557265328.861.2
31936048141.548.5
33364472527.362.7
33637750541.748.3
33652762532.557.5
34142054139.150.9
34880587723.466.6
36462772530.159.9
36846559338.451.6
36980088124.865.2
37292599721.968.1
38555267334.955.1
38788496523.666.4
39640356544.545.5

a
b
c
A in degrees
B in degrees
40056168935.554.5
40762474533.156.9
42085194926.363.7
42946062943.047.0
42970082131.558.5
43266579333.057.0
45178090130.060.0
45552869740.849.2
46477790530.859.2
46859575738.251.8
47386498528.761.3
48160076938.751.3
50470386535.654.4
53375692535.254.8
54062982940.649.4
55557279744.145.9
58074194138.151.9
61572895340.249.8
61666390542.947.1
69669798545.045.0