Maths explained
Pythagoras theorem explained simply, with worked examples
Pythagoras theorem in plain words: what a² + b² = c² means, how to find any missing side, how to test for a right angle, and common mistakes to avoid.
For students. By the Qlive team. . 6 min read.
This guide is for you if you are meeting Pythagoras theorem for the first time or need a clear refresher. In CBSE you first use it around Class 7 and keep using it through Classes 9 and 10. It is also part of ICSE Class 9, Key Stage 3, GCSE and IGCSE maths. Once it clicks, it becomes one of the most useful tools you have in geometry and trigonometry.
The idea in plain words
Pythagoras theorem is a rule about right-angled triangles, triangles with one angle of exactly 90°.
The side opposite the right angle is called the hypotenuse. It is always the longest side. The other two sides are sometimes called the legs.
The theorem says:
a² + b² = c²
where a and b are the two shorter sides and c is the hypotenuse.
In words: if you draw a square on each side of a right-angled triangle, the area of the square on the hypotenuse equals the areas of the other two squares added together.
The classic example is the 3, 4, 5 triangle:
- 3² = 9
- 4² = 16
- 9 + 16 = 25, and 25 = 5²
So a triangle with sides 3, 4 and 5 has a right angle between the sides of length 3 and 4.
The two versions you need
| What you want | Rule | What you do |
|---|---|---|
| The hypotenuse (longest side) | c² = a² + b² | Square both short sides, add, then take the square root |
| A shorter side | a² = c² - b² | Square both known sides, subtract the smaller from the larger, then take the square root |
A quick check: the hypotenuse must come out longer than both other sides. A shorter side must come out shorter than the hypotenuse.
Worked examples
Example 1: finding the hypotenuse
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
- Write the rule: c² = a² + b²
- Substitute: c² = 6² + 8²
- Square: c² = 36 + 64
- Add: c² = 100
- Square root: c = √100 = 10
The hypotenuse is 10 cm. It is longer than 6 and 8, so the answer makes sense.
Example 2: finding a shorter side
The hypotenuse of a right-angled triangle is 13 cm and one shorter side is 5 cm. Find the other side.
- Write the rule for a shorter side: a² = c² - b²
- Substitute: a² = 13² - 5²
- Square: a² = 169 - 25
- Subtract: a² = 144
- Square root: a = √144 = 12
The missing side is 12 cm. It is shorter than 13, so the answer makes sense.
Example 3: a word problem with decimals
A ladder 5 m long leans against a vertical wall. The foot of the ladder is 1.4 m from the wall. How high up the wall does the ladder reach?
First, draw it. The wall and the ground meet at a right angle. The ladder is the slanted side, so the ladder is the hypotenuse. You want a shorter side (the height).
- a² = c² - b²
- a² = 5² - 1.4²
- a² = 25 - 1.96
- a² = 23.04
- a = √23.04 = 4.8
The ladder reaches 4.8 m up the wall. Check: 4.8 × 4.8 = 23.04, and 23.04 + 1.96 = 25 = 5².
Example 4: is it a right-angled triangle?
The theorem also works backwards (this is called the converse). If the two shorter sides squared add up to the longest side squared, the triangle has a right angle.
Triangle with sides 7, 24, 25:
- Longest side: 25, so 25² = 625
- Other two: 7² + 24² = 49 + 576 = 625
- They are equal, so yes, it is right-angled.
Triangle with sides 6, 9, 11:
- Longest side: 11, so 11² = 121
- Other two: 6² + 9² = 36 + 81 = 117
- 117 is not 121, so no, it is not right-angled.
Pythagorean triples worth knowing
Some sets of whole numbers fit the theorem exactly. Spotting them saves time.
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
| 7, 24, 25 | 49 + 576 = 625 |
Any multiple of a triple is also a triple. For example, 6, 8, 10 is 3, 4, 5 doubled.
Common mistakes and how to avoid them
Adding when you should subtract. If the hypotenuse is one of the sides you are given, you are finding a shorter side, so subtract. Always ask first: “Is the side I want the longest one?”
Forgetting the square root. Many students stop at c² = 100 and write 100 as the answer. The last step is always the square root. If your answer is much bigger than the sides you started with, you have probably skipped it.
Square rooting each term separately. √(36 + 64) is √100 = 10. It is not √36 + √64 = 6 + 8 = 14. Add first, then square root.
Using it on a triangle without a right angle. Look for the small square symbol in the corner, or words like “vertical wall” and “flat ground”. If there is no right angle, Pythagoras does not apply.
Picking the wrong side as the hypotenuse. The hypotenuse is opposite the right angle, not just whichever side is drawn at a slant. Mark it first, before you write any numbers.
Rounding too early. Keep the full value until the end, then round to the accuracy the question asks for.
Try these
- The two shorter sides of a right-angled triangle are 9 cm and 12 cm. Find the hypotenuse.
- The hypotenuse is 17 cm and one shorter side is 8 cm. Find the other side.
- A rectangle is 20 cm long and 21 cm wide. Find the length of its diagonal.
- Is a triangle with sides 10 cm, 24 cm and 26 cm right-angled? Show why.
- The shorter sides of a right-angled triangle are 3 cm and 5 cm. Find the hypotenuse, correct to 1 decimal place.
Answers
- c² = 81 + 144 = 225, so c = 15 cm.
- a² = 289 - 64 = 225, so a = 15 cm.
- The diagonal splits the rectangle into two right-angled triangles. d² = 400 + 441 = 841, so d = 29 cm.
- 10² + 24² = 100 + 576 = 676, and 26² = 676. They are equal, so yes.
- c² = 9 + 25 = 34, so c = √34 = 5.830…, which is 5.8 cm to 1 decimal place.
Still unsure?
If Pythagoras still feels shaky, especially in word problems, a Qlive teacher can work through it with you one-to-one at your pace. It starts with a free assessment that finds the exact gap, whether that is the theorem itself or something underneath it like squares and square roots. You can see how we teach the subject on our maths page, and when you are ready for the next step, read our guide to trigonometry and SOHCAHTOA.