Pythagorean Triplets
A Pythagorean triplet is a set of three positive integers (a, b, c) that satisfy the Pythagoras theorem: a² + b² = c².
These are the exact integer side-lengths that can form a right-angled triangle. The most famous example is (3, 4, 5) because 3² + 4² = 9 + 16 = 25 = 5².
Pythagorean triplets have been known for over 4000 years. Ancient Egyptians used the (3, 4, 5) triplet with a knotted rope to construct right angles for building pyramids.
What is Pythagorean Triplets?
Definition: A Pythagorean triplet is a set of three positive integers (a, b, c) such that:
a² + b² = c²
Where:
- a and b are the two shorter sides (legs) of a right-angled triangle
- c is the longest side (hypotenuse)
- All three must be positive integers
Primitive Pythagorean triplet: A triplet where a, b, and c have no common factor other than 1 (i.e., GCD(a, b, c) = 1). Example: (3, 4, 5) is primitive, but (6, 8, 10) is not (it is 2 × (3, 4, 5)).
Common Pythagorean triplets:
- (3, 4, 5)
- (5, 12, 13)
- (8, 15, 17)
- (7, 24, 25)
- (20, 21, 29)
- (9, 40, 41)
Methods
Method 1: General formula for Pythagorean triplets
For any natural number m > 1:
2m, m² − 1, m² + 1
This always generates a Pythagorean triplet because:
- (2m)² + (m² − 1)² = 4m² + m⁴ − 2m² + 1 = m⁴ + 2m² + 1 = (m² + 1)²
Steps:
- Choose any natural number m > 1.
- Calculate: 2m, m² − 1, and m² + 1.
- These three numbers form a Pythagorean triplet.
Method 2: Multiplying a known triplet by a constant
- If (a, b, c) is a Pythagorean triplet, then (ka, kb, kc) is also a Pythagorean triplet for any positive integer k.
- Example: (3, 4, 5) → (6, 8, 10), (9, 12, 15), (12, 16, 20), ...
Method 3: Verification method
- Take three numbers.
- Square each one.
- Check if the sum of the two smaller squares equals the largest square.
Solved Examples
Example 1: Example 1: Generate triplet for m = 2
Problem: Find the Pythagorean triplet using m = 2.
Solution:
Using the formula: 2m, m² − 1, m² + 1
- 2m = 2 × 2 = 4
- m² − 1 = 4 − 1 = 3
- m² + 1 = 4 + 1 = 5
Verification: 3² + 4² = 9 + 16 = 25 = 5². Correct.
Answer: The triplet is (3, 4, 5).
Example 2: Example 2: Generate triplet for m = 3
Problem: Find the Pythagorean triplet using m = 3.
Solution:
Using the formula:
- 2m = 6
- m² − 1 = 9 − 1 = 8
- m² + 1 = 9 + 1 = 10
Verification: 6² + 8² = 36 + 64 = 100 = 10². Correct.
Answer: The triplet is (6, 8, 10).
Example 3: Example 3: Generate triplet for m = 4
Problem: Find the Pythagorean triplet using m = 4.
Solution:
Using the formula:
- 2m = 8
- m² − 1 = 16 − 1 = 15
- m² + 1 = 16 + 1 = 17
Verification: 8² + 15² = 64 + 225 = 289 = 17². Correct.
Answer: The triplet is (8, 15, 17).
Example 4: Example 4: Verify (5, 12, 13)
Problem: Verify that (5, 12, 13) is a Pythagorean triplet.
Solution:
Check: a² + b² = c²?
- 5² + 12² = 25 + 144 = 169
- 13² = 169
- 169 = 169. Yes!
Answer: (5, 12, 13) is a Pythagorean triplet.
Example 5: Example 5: Is (7, 10, 12) a Pythagorean triplet?
Problem: Check if (7, 10, 12) is a Pythagorean triplet.
Solution:
Check: a² + b² = c²?
- 7² + 10² = 49 + 100 = 149
- 12² = 144
- 149 ≠ 144
Answer: No, (7, 10, 12) is NOT a Pythagorean triplet.
Example 6: Example 6: Find the missing number
Problem: (?, 24, 25) is a Pythagorean triplet. Find the missing number.
Solution:
Given:
- b = 24, c = 25 (hypotenuse)
Using a² + b² = c²:
- a² + 24² = 25²
- a² + 576 = 625
- a² = 625 − 576 = 49
- a = √49 = 7
Answer: The triplet is (7, 24, 25).
Example 7: Example 7: Generate triplet for m = 5
Problem: Find the Pythagorean triplet using m = 5.
Solution:
Using the formula:
- 2m = 10
- m² − 1 = 25 − 1 = 24
- m² + 1 = 25 + 1 = 26
Verification: 10² + 24² = 100 + 576 = 676 = 26². Correct.
Answer: The triplet is (10, 24, 26).
Example 8: Example 8: Multiply to create new triplets
Problem: Starting from (3, 4, 5), create 3 more Pythagorean triplets.
Solution:
Multiply by k = 2, 3, 4:
- k = 2: (6, 8, 10). Check: 36 + 64 = 100. Correct.
- k = 3: (9, 12, 15). Check: 81 + 144 = 225. Correct.
- k = 4: (12, 16, 20). Check: 144 + 256 = 400. Correct.
Answer: (6, 8, 10), (9, 12, 15), and (12, 16, 20).
Example 9: Example 9: Application — right triangle sides
Problem: A right-angled triangle has legs of length 9 cm and 40 cm. Find the hypotenuse. Is this a Pythagorean triplet?
Solution:
Given:
- a = 9 cm, b = 40 cm
Using Pythagoras theorem:
- c² = 9² + 40² = 81 + 1600 = 1681
- c = √1681 = 41
Since 9, 40, and 41 are all positive integers, (9, 40, 41) is a Pythagorean triplet.
Answer: Hypotenuse = 41 cm. Yes, it is a Pythagorean triplet.
Example 10: Example 10: Which m generates the triplet containing 20?
Problem: Find the value of m for which the formula 2m, m² − 1, m² + 1 gives a triplet containing 20.
Solution:
Case 1: 2m = 20 → m = 10
- Triplet: 20, 99, 101
- Check: 20² + 99² = 400 + 9801 = 10201 = 101². Correct.
Case 2: m² − 1 = 20 → m² = 21 → m is not an integer. Discard.
Case 3: m² + 1 = 20 → m² = 19 → m is not an integer. Discard.
Answer: m = 10 gives the triplet (20, 99, 101).
Real-World Applications
Real-world applications of Pythagorean triplets:
- Construction: Builders use the (3, 4, 5) triplet to check if corners are right angles. A 3-4-5 rope triangle creates a perfect 90° angle.
- Architecture: Roof trusses, staircases, and ramps use Pythagorean triplets for structural calculations.
- Navigation: Finding the shortest distance between two points (straight-line distance) when only north-south and east-west distances are known.
- Computer graphics: Pixel-perfect diagonal lines and distance calculations use integer triplets to avoid rounding errors.
- Carpentry: Checking if a frame is square (right-angled) by measuring the diagonal.
- Sports: Calculating the distance of a throw or kick using perpendicular components.
Key Points to Remember
- A Pythagorean triplet (a, b, c) satisfies a² + b² = c² where a, b, c are positive integers.
- The formula 2m, m² − 1, m² + 1 generates Pythagorean triplets for any m > 1.
- Multiplying a known triplet by any positive integer k gives another valid triplet.
- Common triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41).
- At least one number in every triplet is divisible by 3 and at least one by 4.
- The product of the three numbers is always divisible by 60.
- There are infinitely many Pythagorean triplets.
- A primitive triplet has GCD = 1. All others are multiples of a primitive triplet.
- Pythagorean triplets give the exact integer sides of a right-angled triangle.
- Not all right triangles have integer sides — only those whose sides form a Pythagorean triplet.
Practice Problems
- Generate the Pythagorean triplet for m = 6.
- Verify whether (11, 60, 61) is a Pythagorean triplet.
- Is (9, 12, 15) a primitive Pythagorean triplet? Why or why not?
- Find the missing number: (?, 40, 41).
- Using the formula, generate 3 different Pythagorean triplets.
- A ladder of length 13 m leans against a wall. The foot is 5 m from the wall. How high does the ladder reach? Identify the Pythagorean triplet.
- Starting from (5, 12, 13), generate two more triplets by multiplying.
- Is (15, 20, 25) a Pythagorean triplet? Is it primitive?
Frequently Asked Questions
Q1. What is a Pythagorean triplet?
A Pythagorean triplet is a set of three positive integers (a, b, c) that satisfy a² + b² = c². These integers form the sides of a right-angled triangle.
Q2. What is the most common Pythagorean triplet?
The most common and simplest Pythagorean triplet is (3, 4, 5). It is used widely in construction and measurement to create right angles.
Q3. What is the formula to generate Pythagorean triplets?
For any natural number m > 1, the numbers 2m, m² − 1, and m² + 1 form a Pythagorean triplet. For example, m = 3 gives (6, 8, 10).
Q4. Are there infinitely many Pythagorean triplets?
Yes. Since the formula works for any m > 1, and there are infinitely many natural numbers, there are infinitely many Pythagorean triplets.
Q5. What is a primitive Pythagorean triplet?
A primitive triplet is one where the three numbers have no common factor other than 1. For example, (3, 4, 5) is primitive, but (6, 8, 10) is not because all three are divisible by 2.
Q6. Can a Pythagorean triplet have all odd numbers?
No. At least one number in every Pythagorean triplet must be even. If a and b were both odd, then a² + b² would be even but not divisible by 4, which means c² would not be a perfect square.
Q7. Does the formula generate all Pythagorean triplets?
The formula 2m, m² − 1, m² + 1 generates many triplets, but not all of them. For instance, (5, 12, 13) requires a more general formula. The complete formula uses two parameters m and n: a = m² − n², b = 2mn, c = m² + n² (where m > n > 0).
Q8. How is the (3, 4, 5) triplet used in construction?
Workers use a rope with 12 equally-spaced knots (3 + 4 + 5 = 12). They form a triangle with sides 3, 4, and 5 units. The corner between the 3-unit and 4-unit sides is exactly 90°.
Q9. How do I find if a number can be part of a Pythagorean triplet?
Use the formula: set 2m = the number (if even) to find m, then compute m² − 1 and m² + 1. If the number is odd, set m² − 1 = the number and solve for m.
Q10. What is the relationship between Pythagorean triplets and the Pythagoras theorem?
The Pythagoras theorem states a² + b² = c² for any right triangle. When a, b, and c happen to be positive integers, they form a Pythagorean triplet. The triplet gives specific integer solutions to the theorem.
