# The Pythagorean theorem

Canonical: https://duckyhelper.com/learn/geometry/pythagorean-theorem/
Updated: 2026-10-01

In a right triangle, the legs a and b and the hypotenuse c (the longest side, across from the right angle) always satisfy \(a^2 + b^2 = c^2\). Use it to find a missing side. The converse tells you a triangle's type from its sides: if \(a^2 + b^2 = c^2\) it is right, if greater it is acute, and if less it is obtuse. The distance formula is the same rule on a graph.

## The key idea

Label the hypotenuse c first. It is always across from the right angle and always the longest side. The other two sides are the legs, a and b.

$$
a^2 + b^2 = c^2 \qquad c = \sqrt{a^2 + b^2} \qquad a = \sqrt{c^2 - b^2}
$$

To find the hypotenuse, add the squares. To find a leg, subtract. Choosing the wrong one of those causes most wrong answers.

### Pythagorean triples

Some sets of whole numbers fit exactly. Any multiple of a triple is also a triple, so 6, 8, 10 and 9, 12, 15 work because 3, 4, 5 does.

| Triple | Check | Some multiples |
| --- | --- | --- |
| 3, 4, 5 | \(9 + 16 = 25\) | 6, 8, 10 and 9, 12, 15 |
| 5, 12, 13 | \(25 + 144 = 169\) | 10, 24, 26 |
| 8, 15, 17 | \(64 + 225 = 289\) | 16, 30, 34 |
| 7, 24, 25 | \(49 + 576 = 625\) | 14, 48, 50 |

### The converse: what kind of triangle is it?

Let c be the longest side and compare \(a^2 + b^2\) with \(c^2\).

| If | The triangle is |
| --- | --- |
| \(a^2 + b^2 = c^2\) | Right |
| \(a^2 + b^2 > c^2\) | Acute (every angle is less than 90°) |
| \(a^2 + b^2 < c^2\) | Obtuse (one angle is more than 90°) |

On a graph, the horizontal gap and the vertical gap between two points are the legs of a right triangle, so the distance between the points is its hypotenuse:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

## Worked examples

**Example 1: find the hypotenuse**

Problem: A right triangle has legs of 9 and 12. Find the hypotenuse.

1. The hypotenuse is the unknown, so add the squares of the legs.

   $$
   9^2 + 12^2 = c^2
   $$
2. Square each leg.

   $$
   81 + 144 = c^2
   $$
3. Add.

   $$
   225 = c^2
   $$
4. Take the square root. A length is positive.

   $$
   c = 15
   $$
5. This is the 3, 4, 5 triple times 3, so you could have spotted it without the work.

Answer: \(c = 15\)

**Example 2: find a leg**

Problem: A 20-foot ladder leans against a wall. Its foot is 6 feet from the base of the wall. How high up the wall does the ladder reach?

1. The wall and the ground meet at a right angle. The ladder is across from it, so the ladder is the hypotenuse. The height h is a leg.

   $$
   6^2 + h^2 = 20^2
   $$
2. Square the numbers.

   $$
   36 + h^2 = 400
   $$
3. Subtract 36 from both sides. Finding a leg means subtracting.

   $$
   h^2 = 364
   $$
4. Take the square root and simplify, since \(364 = 4 \cdot 91\).

   $$
   h = \sqrt{364} = 2\sqrt{91}
   $$
5. On a calculator, \(2\sqrt{91} \approx 19.08\). That is a bit less than 20, which makes sense for a ladder standing almost straight up.

Answer: \(2\sqrt{91}\) feet, about 19.08 feet.

**Example 3: use the converse**

Problem: A triangle has sides of 7, 9 and 12. Is it acute, right or obtuse?

1. The longest side is 12, so it plays the role of c.
2. Add the squares of the two shorter sides.

   $$
   7^2 + 9^2 = 49 + 81 = 130
   $$
3. Square the longest side.

   $$
   12^2 = 144
   $$
4. Compare.

   $$
   130 < 144
   $$
5. The two short sides are too short to meet at a right angle, so the angle across from the side of 12 is wider than 90°.

Answer: Obtuse.

**Example 4 (test-hard): the diagonal of a box**

Problem: A box is 6 inches wide, 8 inches long and 24 inches tall. How long is the longest straight rod that fits inside, from a bottom corner to the opposite top corner?

1. First find the diagonal across the bottom of the box. It is the hypotenuse of a right triangle with legs 6 and 8.

   $$
   \sqrt{6^2 + 8^2} = \sqrt{100} = 10
   $$
2. That bottom diagonal and the 24-inch height form a second right triangle, standing up inside the box. The rod is its hypotenuse.

   $$
   \sqrt{10^2 + 24^2} = \sqrt{676} = 26
   $$
3. Shortcut for any box: \(d = \sqrt{l^2 + w^2 + h^2} = \sqrt{36 + 64 + 576} = 26\).

Answer: 26 inches.

## Common mistakes

- **Adding when you need a leg.** For a leg, subtract: \(a^2 = c^2 - b^2\). Fix: if the hypotenuse is given, you subtract.
- **Calling the wrong side c.** c is always across from the right angle. Fix: find the right-angle mark first and look straight across from it.
- **Thinking \(\sqrt{a^2 + b^2} = a + b\).** Legs 8 and 15 give a hypotenuse of 17, not 23. Square roots do not split over addition.
- **Using it on a triangle with no right angle.** \(a^2 + b^2 = c^2\) only holds for right triangles. For other triangles, use the converse to classify them.
- **Forgetting the last square root.** \(c^2 = 225\) means \(c = 15\), not 225. Fix: ask whether your answer is a length or a squared length.

## Quick methods

> **Tip: Spot a triple first**
>
> Before you square anything, check whether two sides are part of a triple or a multiple of one. Legs of 15 and 20 are 3 and 4 times 5, so the hypotenuse is \(5 \cdot 5 = 25\). This is a real shortcut, and it also catches arithmetic slips.

## Practice

**5 practice questions**

1. A right triangle has legs of 8 and 15. What is the hypotenuse?
   A. \(17\)
   B. \(23\)
   C. \(\sqrt{161}\)
   D. \(289\)

   Answer: \(17\). \(8^2 + 15^2 = 64 + 225 = 289\), and \(\sqrt{289} = 17\). 23 adds the legs, \(\sqrt{161}\) subtracts instead of adding, and 289 forgets the square root.

2. A right triangle has a hypotenuse of 10 and one leg of 4. What is the other leg?
   A. \(6\)
   B. \(2\sqrt{21}\)
   C. \(2\sqrt{29}\)
   D. \(\sqrt{14}\)

   Answer: \(2\sqrt{21}\). \(10^2 - 4^2 = 100 - 16 = 84\), and \(\sqrt{84} = 2\sqrt{21}\), about 9.17. 6 subtracts the sides without squaring, \(2\sqrt{29}\) adds the squares instead of subtracting, and \(\sqrt{14}\) adds the sides without squaring.

3. Which set of side lengths makes a right triangle?
   A. 5, 11, 12
   B. 9, 40, 41
   C. 6, 7, 9
   D. 10, 20, 25

   Answer: 9, 40, 41. \(81 + 1600 = 1681 = 41^2\). For the others, \(25 + 121 = 146\) is not 144, \(36 + 49 = 85\) is not 81, and \(100 + 400 = 500\) is not 625.

4. What is the distance between the points \((-1, 2)\) and \((5, 10)\)?
   A. \(10\)
   B. \(14\)
   C. \(4\sqrt{5}\)
   D. \(100\)

   Answer: \(10\). The horizontal gap is \(5 - (-1) = 6\) and the vertical gap is \(10 - 2 = 8\), so the distance is \(\sqrt{36 + 64} = 10\). 14 adds the gaps, \(4\sqrt{5}\) uses \(5 - 1 = 4\) and drops the negative sign, and 100 forgets the square root.

5. A triangle has sides of 8, 10 and 13. What kind of triangle is it?
   A. Acute
   B. Right
   C. Obtuse
   D. It cannot be a triangle

   Answer: Obtuse. \(8^2 + 10^2 = 164\) and \(13^2 = 169\). Since \(164 < 169\), the triangle is obtuse. It is a real triangle because \(8 + 10 = 18\) is more than 13.

## Frequently asked questions

### Does the Pythagorean theorem work for every triangle?

No. \(a^2 + b^2 = c^2\) is only true for right triangles. For other triangles, comparing \(a^2 + b^2\) with \(c^2\) tells you whether the triangle is acute or obtuse. To find missing sides in those triangles, you will later learn the law of cosines.

### How do I know which side is the hypotenuse?

It is the side across from the right angle, the one that does not touch the small square mark. It is always the longest side. In a word problem it is usually the slanted thing: the ladder, the ramp, the wire or the diagonal.

### Should I memorize Pythagorean triples?

Memorize 3, 4, 5 and 5, 12, 13 at least, and 8, 15, 17 and 7, 24, 25 if you can. Then watch for multiples like 6, 8, 10. Spotting a triple skips the squaring and square root steps and catches arithmetic slips.

### How is the distance formula related to the Pythagorean theorem?

Draw a right triangle with the two points at the ends of the hypotenuse. The legs are the horizontal gap \(x_2 - x_1\) and the vertical gap \(y_2 - y_1\). Put those into \(a^2 + b^2 = c^2\), solve for c, and you have the distance formula.

## Related

- [Special right triangles: 45-45-90 and 30-60-90](https://duckyhelper.com/learn/geometry/special-right-triangles/)
- [How to simplify square roots and radicals](https://duckyhelper.com/learn/algebra-1/simplifying-radicals/)
- [Right triangles and trigonometry on the SAT](https://duckyhelper.com/learn/sat-math/right-triangles-and-trigonometry/)
- [Geometry study guides](https://duckyhelper.com/learn/geometry/)

## Try asking Ducky

- "I got 23 for the hypotenuse with legs 8 and 15. What did I do wrong?"
- "How do I tell which side is c when the triangle is turned sideways?"
- "Give me a ladder problem to try, then check my work."

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