When to Use Pythagoras Theorem (and When to Use Trig)

Quick answer: Use Pythagoras' theorem when a triangle has a right angle and you already know two of its three sides. Then a^2 + b^2 = c^2 gives you the third side, where c is always the hypotenuse. The moment a question hands you an angle, or asks for one, stop — that is trigonometry, not Pythagoras.

I spent most of Year 9 treating Pythagoras as the answer to every triangle. If a shape had three sides, I squared things. Then in a Cambridge IGCSE mock I burned six minutes forcing a^2 + b^2 = c^2 onto a triangle with a 40 degree angle and no right angle anywhere. Zero marks. The fix was not more maths. It was a two-second check.

The two conditions that must both be true

Pythagoras applies only when both are true: the triangle contains a right angle, and you already know two of the three sides. Miss either and the formula is not available.

The right angle is usually a small square in the corner, but often you must infer it — a wall meeting the ground, a ladder against a building, a mast beside its shadow. All are right angles even unlabelled.

The second condition is the one people forget. Two sides. Not one side and one angle. Give me one side and a 35 degree angle and Pythagoras is useless however right-angled the triangle is. The formula contains only sides. There is nowhere to put an angle.

And c is not "the third letter". c is the hypotenuse: the longest side, opposite the right angle, never touching it. I label it with a small H before substituting, every time.

Pythagoras or trigonometry? One question decides it

Ask one thing: does any angle other than the 90 degrees appear in this question, given or wanted?

• No angle mentioned anywhere, so Pythagoras.
• An angle is given, or an angle is the answer, so trigonometry and SOH CAH TOA.
• An angle is involved but there is no right angle at all, so the sine or cosine rule instead.

That is the whole decision. Sides only means Pythagoras. Angles involved means trig. Across GCSE, Cambridge IGCSE, CBSE and WASSCE material I have not met a right-angled triangle question where that test failed.

Plenty of multi-step questions need both: Pythagoras first for a missing side, then trig on that side for an angle. When a question feels stuck, check whether Pythagoras can hand you a number you do not have yet. More of this on the maths hub.

Worked example: finding a shorter side

A ladder 6.5 m long leans against a vertical wall. Its foot is 2.5 m from the base of the wall. How far up the wall does it reach?

1. Spot the right angle. The wall meets the ground at 90 degrees, though nobody says so.
2. Identify the hypotenuse. The ladder is the sloped side, opposite the right angle, so c = 6.5.
3. Choose your direction. I want a shorter side, so I subtract rather than add: a^2 = c^2 - b^2.
4. Substitute. a^2 = 6.5^2 - 2.5^2 = 42.25 - 6.25 = 36.
5. Square root, then units. a = sqrt(36) = 6, so the ladder reaches 6 m up the wall.

The common wrong answer is 6.98 m, from adding when you should subtract. A free check catches it: the hypotenuse must be the biggest length in the triangle, so if your shorter side comes out larger, you added by mistake.

Where Pythagoras hides in disguise

Most exam questions never say the word Pythagoras. Three disguises:

Coordinates. Distance between two points is a Pythagoras question wearing a coat. For A(-2, 1) and B(4, 9), the horizontal difference is 6 and the vertical difference is 8, so AB = sqrt(6^2 + 8^2) = sqrt(100) = 10 units. Sketch the little triangle rather than memorising a distance formula.

Isosceles triangles. A 13 cm, 13 cm, 10 cm triangle is not right-angled, so Pythagoras looks unavailable. Drop a perpendicular from the apex to the middle of the base and you get two right-angled triangles with base 5. Height = sqrt(13^2 - 5^2) = sqrt(144) = 12 cm, so area is 0.5 x 10 x 12 = 60 cm^2. Drawing that perpendicular is worth a method mark on its own.

Cuboids and 3D. The space diagonal of a box 8 cm by 6 cm by 5 cm needs Pythagoras twice. First the base diagonal: sqrt(8^2 + 6^2) = sqrt(100) = 10 cm. Then treat that 10 cm as one side of an upright triangle whose other side is the 5 cm height: sqrt(10^2 + 5^2) = sqrt(125) = 11.2 cm to 3 significant figures. Never round the first diagonal before reusing it: if it came out as sqrt(89), carry sqrt(89) forward, not 9.43.

The converse: proving a triangle is right-angled

Pythagoras also runs backwards, and this is the version students practise least. If a^2 + b^2 = c^2 holds for three given sides, the triangle is right-angled. If not, it is not.

Take 7 cm, 24 cm and 25 cm. 25^2 = 625, and 7^2 + 24^2 = 49 + 576 = 625. They match, so it is right-angled, with the right angle opposite the 25 cm side. Now try 5 cm, 6 cm and 8 cm. 8^2 = 64, but 5^2 + 6^2 = 25 + 36 = 61. Not equal, so not right-angled.

When a question says "show that" or "prove", the mark scheme wants both numbers written out and a conclusion in words. Writing 625 = 625 and stopping often loses the final mark. Add the sentence: since 7^2 + 24^2 = 25^2, the triangle is right-angled.

Where the marks actually go

Most lost marks here are habit errors, not maths errors. Five that cost me most:

• Putting the hypotenuse in the wrong slot. Write 2.5^2 + 6.5^2 in the ladder question and every line after it is wrong.
• Forgetting the square root. You reach 36 and write 36. Write the root sign in the same motion as the subtraction.
• Rounding too early. Keep full calculator accuracy until the final line, then round to 3 significant figures unless told otherwise.
• Dropping units. Metres, centimetres, cm^2 for area. GCSE (AQA, Edexcel, OCR), Cambridge IGCSE, CBSE and WASSCE papers all reserve marks for units.
• Using it on a triangle with no right angle and no way to create one.

For steps on your own questions, the math solver works line by line and explain will re-explain any step. To train the decision rather than the arithmetic, build a mixed set on quiz. Ten Pythagoras questions in a row teach you nothing about when to use it.

Test yourself

1. A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the third side.
2. A question gives a right-angled triangle with one side of 12 m and an angle of 35 degrees, then asks for another side. Pythagoras or trigonometry?
3. Are 9 cm, 12 cm and 15 cm a right-angled triangle? Show why.

Answers: 15 cm, since sqrt(17^2 - 8^2) = sqrt(225). Trigonometry, because an angle is given. Yes, since 9^2 + 12^2 = 225 = 15^2.

And if you are aiming at a specific grade, see the GCSE maths pass mark and boundaries.

FAQ

Does Pythagoras' theorem work on any triangle?

No, only on right-angled triangles. Without a right angle you need the cosine rule, which is really Pythagoras with a correction term bolted on. If a triangle cannot be split into right-angled parts by a perpendicular, a^2 + b^2 = c^2 will hand you a wrong answer very confidently.

How do I know which side is the hypotenuse?

Find the right angle, then look straight across. The side facing it, not touching it, is the hypotenuse, always the longest of the three. Exam diagrams are rarely to scale, so trust the right-angle marker, not which side looks longest.

Do I need to memorise a^2 + b^2 = c^2?

Yes. It is not on the formula sheets for GCSE, Cambridge IGCSE, CBSE or WASSCE papers, and you will use it constantly: trigonometry, coordinate geometry, vectors, later A-Level work.

Can I use Pythagoras in three dimensions?

Yes, by applying it twice. Find a diagonal across a flat face first, then use it as one side of a second right-angled triangle standing upright. Two clear steps are easier to check than one combined formula, and still earn method marks if the arithmetic slips.

In short

Pythagoras is not the default triangle tool, it is a specific one: right angle present, two sides known, no angles in the question. Run that check before you write a number, label the hypotenuse as c, and remember it runs backwards too. Students who lose marks here rarely fail at squaring numbers. They fail at noticing they picked the wrong tool three lines earlier.