Baudhāyana triples
Whole numbers (a, b, c) that fit a right triangle.
Whole-number right triangles
The triangle with sides 3, 4, 5 is a right triangle: 3² + 4² = 5². All three sides are whole numbers.
A set of three whole numbers (a, b, c) that makes a right triangle is called a Baudhāyana triple. Baudhāyana himself listed several. Are they easy to find?
Which of these do you think is a Baudhāyana triple?
What this lesson covers
The idea
A triple of positive integers (a, b, c) that forms the sidelengths of a right triangle, equivalently satisfies a² + b² = c², is called a Baudhāyana (or Pythagorean) triple.
Whole-number right triangles
The triangle with sides 3, 4, 5 is a right triangle: 3² + 4² = 5². All three sides are whole numbers.
A set of three whole numbers (a, b, c) that makes a right triangle is called a Baudhāyana triple. Baudhāyana himself listed several. Are they easy to find?
Which of these do you think is a Baudhāyana triple?
- (4, 5, 6)
- (5, 12, 13)
- (6, 7, 8)
Triple hunt
The toy adds a × a + b × b. If the total is a square number c × c, then (a, b, c) is a triple. Try many pairs.
Triples are rare
A triple of whole numbers (a, b, c) with a² + b² = c² is a Baudhāyana triple. It gives the sides of a right triangle.
Baudhāyana listed (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (12, 35, 37) and (15, 36, 39).
With all three numbers up to 20, there are exactly six: (3, 4, 5), (5, 12, 13), (6, 8, 10), (8, 15, 17), (9, 12, 15), (12, 16, 20).
Notes
A Baudhāyana triple is a triple of whole numbers (a, b, c) with a² + b² = c². It gives the sides of a right triangle. Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).
Check yourself
Which of these is a Baudhāyana triple?
Two perpendicular sides are 20 and 21. The hypotenuse is a whole number. What is it?
Answer: 29
400 + 441 = 841, and 29 × 29 = 841. So (20, 21, 29) is a Baudhāyana triple.
Could (5, 5, c), with two equal sides, be a Baudhāyana triple for a whole number c?
Baudhāyana listed (12, 35, 37). Check it: what is 12² + 35²?
Answer: 1369
144 + 1225 = 1369, and 37 × 37 = 1369. So (12, 35, 37) is a triple.
- (7, 24, 25) — correct. Yes! 7² + 24² = 49 + 576 = 625 = 25².
- (6, 8, 9). 6² + 8² = 36 + 64 = 100, but 9² = 81. Not equal. (6, 8, 10) is the triple.
- (9, 12, 16). 9² + 12² = 81 + 144 = 225, but 16² = 256. Not equal. (9, 12, 15) is the triple.
- Yes, with c = 7.. 5² + 5² = 50, but 7² = 49. It just misses.
- Yes, but only if the numbers are big enough.. Size does not help. With equal sides, c × c = 2 × a × a, so c = a√2.
- No. c would be 5√2, and √2 is not a fraction, so c cannot be a whole number. — correct. Yes! With two equal sides, c = a√2. Since √2 is not a fraction, a√2 is never a whole number.