Pythagoras' theorem states a² + b² = c², where c is the hypotenuse of a right-angled triangle and a, b are the other two sides. Before applying it, run three checks: confirm the triangle is right-angled, identify the hypotenuse correctly, then decide whether to add the two known squares (finding the hypotenuse) or subtract (finding a shorter side). Skipping these checks is the single biggest source of wrong answers, and it also breaks every trig question that depends on this triangle.
Pythagoras' theorem looks like the easiest topic in the syllabus: one formula, three letters. In practice it is where a surprising number of marks disappear, not because students forget the formula, but because they apply it without checking what they are looking at. They subtract when they should add, or they treat the wrong side as the hypotenuse, and the error carries into every step that follows.
This matters beyond Pythagoras itself. Every question in SOH-CAH-TOA and 3D trigonometry assumes you can correctly identify sides in a right-angled triangle first. If that identification is wrong, the sine, cosine, or tangent ratio you build on top of it is wrong too, even if your trig working is flawless.
Check 1
Is there a right angle (marked or given)?
Check 2
Which side is the hypotenuse (opposite the right angle, always the longest)?
Check 3
Am I finding the hypotenuse (add) or a shorter side (subtract)?
Then
Square root your result for the final length
Pythagoras' theorem only works on right-angled triangles. If the triangle has no right angle marked, you cannot use it directly. This is worth stating plainly because in mixed exam questions, students sometimes apply a² + b² = c² to a triangle that is not right-angled simply because two side lengths are given. (If all three sides are given and no right angle is marked, you can test for one using the converse, covered in the FAQ below.)
A right-angled triangle has the two shorter sides 5 cm and 12 cm. Find the length of the hypotenuse.
Solution
c.c² = 5² + 12².c² = 25 + 144 = 169.c = √169 = 13 cm.Quick check: 5, 12, 13 is a common Pythagorean triple. Recognising it (along with 3, 4, 5 and 8, 15, 17) lets you skip the calculator entirely on some questions and catch arithmetic slips instantly.
A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the length of the other shorter side.
Solution
a, is one of the two shorter sides.a² = 17² − 8².a² = 289 − 64 = 225.a = √225 = 15 cm.The step students get wrong
When finding a shorter side, students often add the two given squares instead of subtracting, because "add the squares" from finding the hypotenuse becomes a reflex. The rule is: the hypotenuse squared always equals the sum of the other two squared, so if you already know the hypotenuse, you must subtract the square of the known shorter side to isolate the unknown one. Always confirm which side is missing before deciding whether to add or subtract.
A ladder leans against a wall. The foot of the ladder is 2.4 m from the wall, and the ladder is 6 m long. How high up the wall does the ladder reach?
Solution
h² = 6² − 2.4².h² = 36 − 5.76 = 30.24.h = √30.24 ≈ 5.50 m (to 3 significant figures).Exam tip: word problems like ladders, ramps, and diagonal supports almost always hide a right-angled triangle. Sketch the triangle first, label the hypotenuse and the two shorter sides clearly, then run the 3-step check before you write any formula.
In SOH-CAH-TOA, you choose sine, cosine, or tangent based on which two sides of a right-angled triangle you know or want. Getting the hypotenuse wrong there does not just cost the Pythagoras mark, it flips your entire ratio. In 3D trigonometry, the first move on most questions is to slice a 3D solid into a flat right-angled triangle using Pythagoras, before any angle is found. If that first triangle is set up incorrectly, every angle calculated afterward is built on a wrong length.
It also turns up outside trigonometry. The distance formula in coordinate geometry is Pythagoras' theorem applied to the horizontal and vertical gaps between two points, so the same hypotenuse thinking carries straight over.
Treat the 3-step check as a habit, not an extra step: right angle confirmed, hypotenuse identified, add or subtract decided. Doing this automatically before you write any working is what prevents a small identification error from becoming a wrong final answer three steps later.
After calculating, use your calculator to verify the squared relationship holds, rather than trusting a single pass of arithmetic.
On the fx-97SG CW (ClassWiz) and fx-97SG X
x² key, and add or subtract them exactly as your 3-step check decided.a² + b² = c² with all three sides to confirm both sides of the equation match.Does Pythagoras' theorem work on any triangle?
No. It only works on right-angled triangles. If a triangle has no right angle, you need the sine rule or cosine rule instead, not Pythagoras' theorem.
What if the question asks me to show a triangle is right-angled?
Use the theorem in reverse (the converse). Square all three given sides, then test whether the largest square equals the sum of the other two. For a triangle with sides 9 cm, 40 cm and 41 cm: 9² + 40² = 81 + 1600 = 1681 and 41² = 1681, so the two match and the triangle is right-angled, with the right angle opposite the 41 cm side. If the two values do not match, the triangle is not right-angled and Pythagoras' theorem cannot be used on it.
How do I know which side is the hypotenuse?
The hypotenuse is always the side directly opposite the right angle, and it is always the longest side of the triangle. If you are unsure, sketch the triangle, mark the right angle clearly, and trace the side facing it.
Why do I sometimes add and sometimes subtract?
You add the two known squared values only when you are finding the hypotenuse. If you already know the hypotenuse and are finding one of the shorter sides, you subtract the known shorter side's square from the hypotenuse's square. Deciding this correctly is exactly what the 3-step check is for.
Can the answer be a decimal, not a whole number?
Yes. Only specific combinations, called Pythagorean triples such as 3, 4, 5 or 5, 12, 13, give whole-number answers. Most exam questions will not, so expect to round your final answer to the number of significant figures the question asks for.
— Mr Gan Math Tuition
Mr. Gan builds the 3-step check into a habit so it carries straight into SOH-CAH-TOA and 3D trigonometry questions.
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