Pythagoras’ Theorem
One formula, used constantly — inside trigonometry, mensuration, coordinate geometry and vectors. The whole topic reduces to knowing which side is the hypotenuse and therefore whether to add or subtract.
1. The theorem
a² + b² = c², where c is the HYPOTENUSE
The hypotenuse is the side opposite the right angle, and always the longest side.
This only works in a RIGHT-ANGLED triangle. Without a right angle you need the cosine rule instead.
Identify the hypotenuse before writing anything. Putting a shorter side where c belongs was the most frequent error recorded — and it produces an answer that looks reasonable.
2. Finding the hypotenuse — ADD
Example: shorter sides 6 cm and 8 cm.
- c² = 6² + 8²
- c² = 36 + 64 = 100
- c = √100 = 10 cm
The hypotenuse must come out LONGER than both other sides. If it doesn’t, you’ve made an error — this is the fastest check available.
3. Finding a shorter side — SUBTRACT
When the hypotenuse is known and a shorter side is missing, rearrange:
a² = c² − b²
Example: hypotenuse 13 cm, one shorter side 5 cm.
- a² = 13² − 5²
- a² = 169 − 25 = 144
- a = √144 = 12 cm
Subtract when finding a shorter side. Adding the squares was recorded repeatedly and gives an impossible answer — a shorter side coming out longer than the hypotenuse.
Square the sides first, then subtract. 13² − 5² = 169 − 25 = 144. It is not (13 − 5)² = 64.
The decision in one line:
| Missing side | Operation |
|---|---|
| Hypotenuse (the longest) | ADD the squares |
| A shorter side | SUBTRACT from the hypotenuse squared |
A negative result means you subtracted the wrong way round — you have treated a shorter side as the hypotenuse.
4. Method and accuracy
- Sketch the triangle and mark the right angle
- Label the hypotenuse
- Decide add or subtract
- Substitute, then square root
- Check the answer is sensible
Take the positive square root. Algebraically x² = 144 gives x = ±12, but a length cannot be negative, so the answer is 12. Say so if the question involves solving.
Don’t round intermediate values. √130 is 11.4018…, not 11.3 — carry the full value if it feeds into another step, and round only at the end.
Give 3 significant figures unless told otherwise, and include units.
Check the units match before calculating — mixing km and cm was a documented error.
5. Common applications
Isosceles triangles
Drop a perpendicular from the apex to the base. It bisects the base, creating two right-angled triangles.
The perpendicular halves the base. For a base of 30, each half is 15 — using the full 30 was a recorded error. This construction is how you find the height of an isosceles triangle, and hence its area.
Composite shapes
Break the shape into right-angled triangles and apply Pythagoras step by step, carrying full accuracy between stages.
Coordinate geometry
The distance between two points is Pythagoras in disguise:
length = √[(x₂−x₁)² + (y₂−y₁)²]
Vectors
The magnitude of a column vector: |a| = √(x² + y²).
Cones
The slant height, radius and perpendicular height form a right-angled triangle:
l² = r² + h²
3D problems
Apply Pythagoras twice — once in the base, then in the vertical triangle. For a cuboid’s space diagonal:
d² = l² + w² + h²
6. Is it right-angled?
Pythagoras works in reverse as a test:
If a² + b² = c² (with c the longest side), the triangle is right-angled.
Example: 5, 12, 13 → 25 + 144 = 169 = 13² ✓ → right-angled.
Useful triples to recognise: 3-4-5, 5-12-13, 8-15-17, and their multiples (6-8-10, 9-12-15).
7. Pythagoras or trigonometry?
| You have | Use |
|---|---|
| Three sides (two known, one missing), no angles | Pythagoras |
| Sides and an angle | Trigonometry (SOHCAHTOA) |
| Want to find an angle | Trigonometry — never Pythagoras |
| No right angle | Sine or cosine rule |
Pythagoras cannot find an angle. It relates the three sides only — a misconception that appeared in lessons more than once.
8. Mistakes that cost marks
Misidentifying the hypotenuse.
Adding when you should subtract for a shorter side.
Computing (a − b)² instead of a² − b².
Using Pythagoras in a triangle with no right angle.
Trying to find an angle with Pythagoras.
Forgetting to square root at the end.
Giving a negative length.
Using the whole base instead of half in an isosceles triangle.
Mixing units.
Rounding too early.
Omitting units or the required accuracy.
Frequently asked questions
What is Pythagoras’ theorem? a² + b² = c², where c is the hypotenuse of a right-angled triangle.
Which side is the hypotenuse? The one opposite the right angle — always the longest.
When do I add and when do I subtract? Add to find the hypotenuse; subtract to find a shorter side.
What if I get a negative number? You have subtracted the wrong way round — check which side is the hypotenuse.
Can Pythagoras find an angle? No — use trigonometry for angles.
Does it work in any triangle? Only in right-angled triangles. Otherwise use the cosine rule.
How do I use it in an isosceles triangle? Drop a perpendicular from the apex; it bisects the base, giving a right-angled triangle.
How do I check if a triangle is right-angled? Test whether a² + b² = c² with c the longest side.
How do I use Pythagoras in 3D? Apply it twice, or use d² = l² + w² + h² for a cuboid’s diagonal.
Quick revision checklist
- I can identify the hypotenuse every time
- I know a² + b² = c² applies only to right-angled triangles
- I add to find the hypotenuse
- I subtract to find a shorter side
- I square before subtracting
- I check the hypotenuse is the longest side
- I take the positive square root for a length
- I can use it in isosceles triangles by halving the base
- I can break composite shapes into right-angled triangles
- I recognise 3-4-5, 5-12-13 and 8-15-17
- I can test whether a triangle is right-angled
- I can apply it in 3D
- I know to use trigonometry for angles
- I carry full accuracy and give 3 s.f. with units
These notes cover Pythagoras’ theorem in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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