Pythagorean Triples -- List, Formula, and How to Find Them

Grade: 8-9 | Topic: Geometry

What You Will Learn

After this lesson you will know what Pythagorean triples are, recognize the most common ones by sight, understand the difference between primitive and non-primitive triples, and be able to generate new triples using a formula. Recognizing triples saves time on tests because you can identify right-triangle side lengths instantly without calculating.

Theory

What is a Pythagorean triple?

A Pythagorean triple is a set of three positive whole numbers $(a, b, c)$ where:

$a^2 + b^2 = c^2$

The most famous example is (3, 4, 5):

$3^2 + 4^2 = 9 + 16 = 25 = 5^2 \checkmark$

Because all three numbers are integers, these right triangles have "clean" side lengths with no square roots or decimals.

Common Pythagorean triples

Memorizing these triples is extremely useful for quick problem-solving:

Triple	Verification
(3, 4, 5)	$9 + 16 = 25$
(5, 12, 13)	$25 + 144 = 169$
(8, 15, 17)	$64 + 225 = 289$
(7, 24, 25)	$49 + 576 = 625$
(9, 40, 41)	$81 + 1600 = 1681$
(11, 60, 61)	$121 + 3600 = 3721$
(20, 21, 29)	$400 + 441 = 841$

Scaling triples (non-primitive triples)

If $(a, b, c)$ is a Pythagorean triple, then multiplying all three numbers by the same positive integer $k$ gives another triple:

$(ka)^2 + (kb)^2 = k^2 a^2 + k^2 b^2 = k^2(a^2 + b^2) = k^2 c^2 = (kc)^2$

For example, starting with (3, 4, 5):

Scale factor $k$	Triple
2	(6, 8, 10)
3	(9, 12, 15)
4	(12, 16, 20)
5	(15, 20, 25)
10	(30, 40, 50)

A primitive triple is one where $\gcd(a, b, c) = 1$ -- that is, the three numbers share no common factor greater than 1. Every non-primitive triple is a scaled version of some primitive triple.

The generating formula

There is a formula that produces every primitive Pythagorean triple. Choose two positive integers $m$ and $n$ where $m > n$, they have opposite parity (one even, one odd), and $\gcd(m, n) = 1$. Then:

$a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2$

Let's verify with $m = 2$, $n = 1$:

$a = 4 - 1 = 3, \quad b = 2 \times 2 \times 1 = 4, \quad c = 4 + 1 = 5$

That gives us (3, 4, 5). Here are a few more:

$m$	$n$	$a = m^2 - n^2$	$b = 2mn$	$c = m^2 + n^2$	Triple
2	1	3	4	5	(3, 4, 5)
3	2	5	12	13	(5, 12, 13)
4	1	15	8	17	(8, 15, 17)
4	3	7	24	25	(7, 24, 25)
5	2	21	20	29	(20, 21, 29)
5	4	9	40	41	(9, 40, 41)

Why the formula works

Substitute into the Pythagorean equation:

a^2 + b^2 &= (m^2 - n^2)^2 + (2mn)^2 \\ &= m^4 - 2m^2n^2 + n^4 + 4m^2n^2 \\ &= m^4 + 2m^2n^2 + n^4 \\ &= (m^2 + n^2)^2 \\ &= c^2 \end{aligned}$$ The algebra confirms that $a^2 + b^2 = c^2$ for any choice of $m$ and $n$. ## Worked Examples ### Example 1: Recognizing a scaled triple (easy) **Problem:** A right triangle has legs 9 and 12. Find the hypotenuse without using the formula. **Step 1:** Check whether the legs match a known triple scaled up. Divide both by 3: $$9 \div 3 = 3, \quad 12 \div 3 = 4$$ This is the (3, 4, 5) triple scaled by 3. **Step 2:** The hypotenuse is: $$c = 5 \times 3 = 15$$ **Answer:** The hypotenuse is **15**. (Verify: $9^2 + 12^2 = 81 + 144 = 225 = 15^2$.) ### Example 2: Checking if three numbers form a triple (easy) **Problem:** Is (10, 24, 26) a Pythagorean triple? **Step 1:** Check the equation. $$10^2 + 24^2 = 100 + 576 = 676$$ $$26^2 = 676$$ **Step 2:** $676 = 676$ -- yes. **Step 3:** Is it primitive? $\gcd(10, 24, 26) = 2$. Since they share a common factor of 2, this is **not** primitive. It is 2 times the primitive triple (5, 12, 13). **Answer:** Yes, (10, 24, 26) **is** a Pythagorean triple, but it is non-primitive (it equals $2 \times (5, 12, 13)$). ### Example 3: Generating a triple from the formula (medium) **Problem:** Use $m = 6$ and $n = 1$ to generate a Pythagorean triple. **Step 1:** Check the conditions: $m > n$ (yes), opposite parity (6 is even, 1 is odd -- yes), $\gcd(6, 1) = 1$ (yes). **Step 2:** Apply the formula. $$a = m^2 - n^2 = 36 - 1 = 35$$ $$b = 2mn = 2 \times 6 \times 1 = 12$$ $$c = m^2 + n^2 = 36 + 1 = 37$$ **Step 3:** Verify. $$35^2 + 12^2 = 1225 + 144 = 1369 = 37^2 \checkmark$$ **Answer:** The triple is **(12, 35, 37)**. ### Example 4: Finding the missing number in a triple (medium) **Problem:** Two sides of a Pythagorean triple are 20 and 29. Find the third side. **Step 1:** Since 29 > 20, $c = 29$ might be the hypotenuse. Find the missing leg: $$a = \sqrt{29^2 - 20^2} = \sqrt{841 - 400} = \sqrt{441} = 21$$ **Step 2:** Check that 21 is a whole number. Yes. So the triple is (20, 21, 29). **Answer:** The third side is **21**. (This is a well-known primitive triple.) ### Example 5: Is 50-length the hypotenuse of an integer right triangle? (challenging) **Problem:** Find all primitive Pythagorean triples where $c = 65$. **Step 1:** We need $m^2 + n^2 = 65$ with $m > n > 0$, opposite parity, $\gcd(m, n) = 1$. **Step 2:** Find integer solutions to $m^2 + n^2 = 65$: - $m = 8, n = 1$: $64 + 1 = 65$ -- opposite parity, $\gcd = 1$. Valid. - $m = 7, n = 4$: $49 + 16 = 65$ -- opposite parity, $\gcd = 1$. Valid. - $m = 4, n = 7$: invalid since $m$ must be greater than $n$. **Step 3:** Generate the triples. For $m = 8, n = 1$: $$a = 64 - 1 = 63, \quad b = 16, \quad c = 65 \quad \Rightarrow (16, 63, 65)$$ For $m = 7, n = 4$: $$a = 49 - 16 = 33, \quad b = 56, \quad c = 65 \quad \Rightarrow (33, 56, 65)$$ **Step 4:** Verify both: $$16^2 + 63^2 = 256 + 3969 = 4225 = 65^2 \checkmark$$ $$33^2 + 56^2 = 1089 + 3136 = 4225 = 65^2 \checkmark$$ **Answer:** There are two primitive triples with hypotenuse 65: **(16, 63, 65)** and **(33, 56, 65)**. ## Common Mistakes **Mistake 1: Assuming all multiples of a triple are primitive** ❌ Calling (6, 8, 10) a primitive Pythagorean triple. ✅ (6, 8, 10) is a valid triple, but it is **not** primitive because all three share a factor of 2. The underlying primitive triple is (3, 4, 5). Why this matters: The distinction is important when using the generating formula, which only produces primitive triples directly. Knowing a triple is non-primitive helps you simplify problems by working with the smaller base triple. **Mistake 2: Mixing up which number is the hypotenuse when scaling** ❌ Seeing sides 15 and 20 and guessing the third side is 7 (trying to match a 7-24-25 pattern). ✅ Divide by the common factor: $15 \div 5 = 3$, $20 \div 5 = 4$. This is the (3, 4, 5) triple scaled by 5, so $c = 25$. Why this matters: Always reduce to the base triple first. Guessing the wrong family of triples leads to a wrong answer. **Mistake 3: Forgetting the conditions on m and n in the generating formula** ❌ Using $m = 4, n = 2$: $a = 16 - 4 = 12$, $b = 16$, $c = 20$. This gives (12, 16, 20), but $\gcd = 4$, so it is not primitive. ✅ The conditions require $\gcd(m, n) = 1$ and opposite parity. Here $\gcd(4, 2) = 2$ and both are even, so the conditions are violated. Why this matters: Skipping the conditions produces non-primitive triples (or duplicates). The formula only guarantees primitive triples when both conditions are met. ## Practice Problems Try these on your own before checking the answers: 1. Is (14, 48, 50) a Pythagorean triple? If so, is it primitive? 2. A right triangle has legs 36 and 77. Use your knowledge of triples to find the hypotenuse. 3. Use $m = 5, n = 2$ to generate a Pythagorean triple. 4. The sides 28 and 45 are two sides of a Pythagorean triple. Find the third side. 5. List all primitive Pythagorean triples with $c < 30$. <details> <summary>Click to see answers</summary> 1. Check: $14^2 + 48^2 = 196 + 2304 = 2500 = 50^2$. Yes, it is a triple. $\gcd(14, 48, 50) = 2$, so it is **not** primitive. It is $2 \times (7, 24, 25)$. 2. Check for common factors: $\gcd(36, 77) = 1$, so this could be primitive. Check: $36^2 + 77^2 = 1296 + 5929 = 7225 = 85^2$. The hypotenuse is **85**. 3. Check conditions: $m > n$ (yes), opposite parity (5 odd, 2 even -- yes), $\gcd(5,2) = 1$ (yes). $a = 25 - 4 = 21$, $b = 20$, $c = 29$. Triple: **(20, 21, 29)**. Verify: $400 + 441 = 841 = 29^2$. 4. Try $c = 53$: $53^2 - 28^2 = 2809 - 784 = 2025 = 45^2$. Yes. Or try $c = 45$, $a = 28$: $45^2 - 28^2 = 2025 - 784 = 1241$, which is not a perfect square. So the third side is the hypotenuse: **53**. 5. Primitive triples with $c < 30$: **(3, 4, 5)**, **(5, 12, 13)**, **(8, 15, 17)**, **(7, 24, 25)**, **(20, 21, 29)**. </details> ## Summary - A **Pythagorean triple** is three positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$. - The most common triples to memorize are (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). - Multiplying all three numbers by the same factor $k$ produces a new (non-primitive) triple. - The **generating formula** $a = m^2 - n^2$, $b = 2mn$, $c = m^2 + n^2$ produces every primitive triple when $m > n$, $\gcd(m, n) = 1$, and $m$, $n$ have opposite parity. - Recognizing triples speeds up problem-solving because you can find missing sides instantly. ## Related Topics - [Pythagorean Theorem -- Formula, Proof, and Examples](/learn/pythagorean-theorem) - [Pythagorean Theorem Examples with Step-by-Step Solutions](/learn/pythagorean-theorem-examples) - [Distance Formula -- How to Find Distance Between Two Points](/learn/distance-formula) --- <div className="mp-cta-box"> **Need help with Pythagorean triples?** Take a photo of your math problem and MathPal will solve it step by step. 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