Right-angled triangles turn up in diagrams, craft projects and exam questions, and Pythagoras' theorem is the tool that links their three sides. The idea is short enough to remember, but small slips — labelling the hypotenuse incorrectly or subtracting when you should add — cause most lost marks. The method below keeps the steps visible so you can find a missing side with confidence.
What Pythagoras' theorem says
For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In symbols:
a² + b² = c²
The letters a and b stand for the two shorter sides that meet at the right angle. The letter c stands for the hypotenuse, the side opposite the right angle. It is always the longest side.
The word “square” is literal. If you drew a square on each side of the triangle, the two smaller squares would together have the same area as the largest square. That is why the theorem uses squares rather than simply adding the side lengths.
How to identify the hypotenuse
The hypotenuse is not simply the side that looks longest, although it will be. It is the side opposite the right angle. Look for the small square symbol in the corner. The side that does not touch that corner is the hypotenuse. In a triangle drawn with a horizontal base and a vertical height, the hypotenuse is the sloping side. But the triangle can be rotated, so use the right angle, not the page.
A quick labelling habit
Before doing any calculation, copy or redraw the triangle. Mark the right angle. Label the hypotenuse c and the two shorter sides a and b. If the question already uses different letters, rename them in your working or adjust the formula. Clear labels prevent the most common error: putting a shorter side into the c² position.
- Right angle: mark it with a square.
- Hypotenuse: opposite the right angle, label it c.
- Shorter sides: the two sides forming the right angle, label them a and b.
Finding the hypotenuse: a worked example
Suppose a right-angled triangle has shorter sides of 6 cm and 8 cm. The hypotenuse is unknown. Write the formula, then substitute the values you know.
c² = a² + b²
c² = 6² + 8²
c² = 36 + 64
c² = 100
Because c² is 100, c is the square root of 100:
c = √100 = 10 cm
Notice that the answer is not 6 + 8 = 14. The theorem works with squares, not with the side lengths themselves. Also, do not stop at 100. 100 is c², not c.
Finding a shorter side: work backwards
If the hypotenuse is known, the calculation changes. Example: the hypotenuse is 13 cm, one shorter side is 5 cm, and you need the other shorter side. Label the unknown shorter side a.
a² + 5² = 13²
a² + 25 = 169
To isolate a², subtract 25 from both sides:
a² = 169 − 25 = 144
a = √144 = 12 cm
Here the key is subtraction. The hypotenuse square is the total, so the square of the unknown shorter side is the difference between 169 and 25. Adding would give 194, which is too large and cannot be right because the missing side is shorter than the hypotenuse. You can check your answer:
5² + 12² = 25 + 144 = 169 = 13²
Squares and square roots without panic
Squaring a number means multiplying it by itself: 7² = 7 × 7 = 49. Finding a square root reverses that: √49 = 7. Most KS3 questions use square numbers that give whole-number answers, but sometimes you will need a calculator and a sensible rounded answer.
Useful square numbers to recognise:
- 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
- 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100
- 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
If the square root is not exact, round only at the end. For example, if c² = 50, then c = √50 ≈ 7.07. If the question asks for one decimal place, write 7.1. Keep the units: cm, m or mm. Square units belong to area, not side lengths.
Mistakes that cause most confusion
Most errors come from a small set of habits. Watch for these:
- Using the theorem on any triangle. It only works when one angle is exactly 90°.
- Calling the longest-looking side the hypotenuse. Find the right angle first; the hypotenuse is opposite it.
- Adding when finding a shorter side. If the hypotenuse is known, subtract the known shorter side's square.
- Forgetting the square root. If you have c² = 100, the side length is 10, not 100.
- Mixing up the order. The sum of the two shorter sides squared equals the hypotenuse squared: a² + b² = c².
- Rounding too early. Carry the full value on your calculator, then round the final answer.
A reliable routine
- Redraw the triangle and mark the right angle.
- Label the hypotenuse c and the shorter sides a and b.
- Write a² + b² = c².
- Substitute the two known values.
- Add if c is missing; subtract if a shorter side is missing.
- Take the square root and include the units.
Checking your answer and using the theorem well
Always sanity-check. The hypotenuse must be longer than each shorter side. If you find a shorter side longer than the hypotenuse, you have subtracted the wrong way or mislabelled the triangle. A quick substitution back into a² + b² = c² will confirm your result.
Some triples are worth recognising: 3, 4, 5; 5, 12, 13; 8, 15, 17; and 7, 24, 25. They are not shortcuts that replace the method, but they help you spot when an answer is likely to be correct. If a question gives 9 and 12, for instance, the hypotenuse is 15 because 9² + 12² = 81 + 144 = 225.
For KS3, the goal is not speed. It is a calm, repeatable process: identify the right angle, label carefully, square, add or subtract, then square-root. Practise with a mixture of missing-hypotenuse and missing-shorter-side questions. Draw the triangle every time until the labels feel automatic. That way, Pythagoras' theorem becomes a reliable tool rather than a formula to memorise in a hurry.
Photo: 422737 / Pixabay