The most famous equation in school
A right-angled triangle is a triangle with one right angle — a corner like a sheet of paper. The two sides that make that corner are the shorter sides. The third side, the longest one opposite the right angle, is the hypotenuse.
And now the theorem. 2500 years ago the Greeks noticed: if you build a square on each side, the areas of the squares on the shorter sides add up to exactly the area of the square on the hypotenuse.
Written with numbers: a² + b² = c², where a, b are the shorter sides and c is the hypotenuse.
What is it for? You know two sides and you calculate the third — without measuring. A wall, a ladder, the diagonal of a pitch, a phone screen. Everywhere a right angle is hiding.
Today you will understand the theorem and calculate your first sides with it. Squares and square roots from the last lessons finally get to show off.
Tereza and Filip are stretching a net across the school pitch. They need a rope from one corner to the opposite corner. The pitch measures 12 m by 9 m. Filip wants to measure the rope by walking. Tereza draws in her notebook: “That is a right-angled triangle. Shorter sides 12 and 9, the rope is the hypotenuse.” She calculates: 12² + 9² = 144 + 81 = 225. The square root of 225 is 15. “Fifteen metres. Buy the rope and you do not have to walk anywhere.” Filip then stretches the rope — and it measures exactly 15 m. A theorem 2500 years old works on the pitch too.
First always get clear which side is the hypotenuse. It is the longest side and it lies opposite the right angle. In the formula it belongs as c — alone on one side of the equals sign.
How to find a missing side
You are looking for the hypotenuse (the longest side):
- Square both shorter sides: a², b².
- Add: a² + b².
- Take the square root: c = √(a² + b²).
Example: shorter sides 3 and 4 → 9 + 16 = 25 → c = √25 = 5.
You are looking for a shorter side (you know the hypotenuse and one shorter side):
- Square the hypotenuse and the known shorter side.
- Subtract: c² − a².
- Take the square root: b = √(c² − a²).
Example: hypotenuse 13, shorter side 5 → 169 − 25 = 144 → b = √144 = 12.
Remember the logic instead of two formulae: the square on the hypotenuse is the sum, so towards the hypotenuse you add and away from the hypotenuse you subtract.
The last step is always a square root — from c² back to c. That is what lesson 2 is for. And always write the unit: sides in metres, areas of squares in m².
Example 1: hypotenuse from shorter sides 6 and 8
Shorter sides a = 6 cm, b = 8 cm. I am looking for hypotenuse c.
c² = a² + b² = 6² + 8² = 36 + 64 = 100.
c = √100 = 10 cm.
Sense check: the hypotenuse must be the longest side. 10 > 8 > 6 — it fits. The triple 6, 8, 10 is, by the way, just a doubled famous triple 3, 4, 5.
Example 2: a shorter side from hypotenuse 17 and shorter side 15
Hypotenuse c = 17 m, shorter side a = 15 m. I am looking for the other shorter side b.
b² = c² − a² = 17² − 15² = 289 − 225 = 64.
b = √64 = 8 m.
Check back by adding: 15² + 8² = 225 + 64 = 289 = 17². It fits. If a shorter side came out longer than the hypotenuse, I would know at once that I added instead of subtracting.
Example 3: is the triangle right-angled?
A triangle has sides 5, 12 and 13 cm. Is it right-angled?
The theorem also works backwards: if a² + b² = c², the triangle is right-angled.
The longest side is 13, that would be the hypotenuse. I try: 5² + 12² = 25 + 144 = 169. And 13² = 169.
They are equal — the triangle is right-angled. Builders check right angles this way: they stretch strings of 3 m, 4 m and 5 m, and if they fit, the corner is right.
The theorem works only for right-angled triangles. If there is no right angle in the question (not even a hidden one — say in a rectangle), Pythagoras will not help. And watch: the hypotenuse is always c, the longest side. Putting it in as a shorter side is the most common year-8 mistake.
Play it
Before you go to the sheet, try it with your eyes. This is not a test — it is a game. Tap until you see it.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.
Find the missing side. You can see the triangle at the top.
On paper
Draw the triangles. A sketch saves more marks than a calculator.
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Draw a right-angled triangle, mark the right angle and label the sides a, b, c. Write the theorem a² + b² = c² next to it.
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The shorter sides measure 9 cm and 12 cm. Calculate the hypotenuse with the full method: squares, sum, square root.
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The hypotenuse measures 25 m, one shorter side 24 m. Calculate the other shorter side and check by adding.
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Check by calculating whether a triangle with sides 8, 15, 17 is right-angled.
Now you 💪
- I can spot the hypotenuse: the longest side opposite the right angle.
- Towards the hypotenuse I add the squares, towards a shorter side I subtract — and I know why.
- I do not forget the last step: the square root.
Done when: You have a labelled triangle drawn, a hypotenuse and a shorter side calculated with working, and a run of five correct in practice.
What to take from this lesson
- In a right-angled triangle a² + b² = c², where c is the hypotenuse.
- Hypotenuse: add the squares of the shorter sides and take the square root.
- Shorter side: subtract the squares and take the square root.
- The theorem also works as a test for a right angle (the triple 3, 4, 5).
- It only works for right-angled triangles.
© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten