vmbo havo vwo
Pythagorean theorem calculator
Quick answer
In a right triangle, a² + b² = c². With a = 3 and b = 4, c = √(9 + 16) = √25 = 5.
A missing leg: b = √(c² − a²). c must be longer than a.
Calculator
Choose hypotenuse or leg. The unit (cm, m, mm) appears only on the answer; we do not convert between units.
What is the Pythagorean theorem?
Dutch school books usually say Pythagoras for the same rule English calls the Pythagorean theorem. It applies only in a triangle with a right angle. The two sides that form that angle are the legs (a and b). The side opposite the right angle is the hypotenuse (c) — always the longest.
| You need | Formula | Watch |
|---|---|---|
| Hypotenuse c | c = √(a² + b²) | Same length unit for a and b (cm, m, …) |
| Leg b | b = √(c² − a²) | c must be strictly longer than a |
On vmbo (the more practical Dutch track), havo and vwo (middle and university-preparatory tracks) this sits in geometry. Later years reuse the same formula for vectors and for the distance between two points. Triangle area is a different tool — see the area calculator; that is not Pythagoras.
Examples by level
A 13 m ladder stands 5 m from the wall. How high does it reach?
- Theorema² + b² = c²; here c = 13 m and a = 5 m
- Squares13² − 5² = 169 − 25 = 144
- Square root√144 = 12
The ladder reaches 12 m. Keep meters to the end; we do not convert to cm.
A square has side 1 cm. How long is the diagonal?
- Legsa = 1, b = 1
- Sum of squares1² + 1² = 2
- Square rootc = √2 ≈ 1.4142 cm
Leave √2 if an exact answer is required. In the calculator we round to 4 decimal places.
Points A(1, 2) and B(4, 6). What is the distance AB?
- DifferencesΔx = 3, Δy = 4
- PythagorasAB = √(3² + 4²) = √(9 + 16) = √25 = 5
The distance formula is Pythagoras in the coordinate plane.
Common mistakes
Wrong
c = a + b, so 3 + 4 = 7.
Right
Square first: 9 + 16 = 25, then the root. c = 5.
Wrong
Finding a leg with √(a² + c²).
Right
Subtract the known leg from c²: √(c² − a²).
Wrong
Using Pythagoras in a triangle with no right angle.
Right
Only with a right angle. Otherwise a² + b² = c² does not hold.
Wrong
Mixing a in cm and b in m without converting.
Right
Same unit first. The tool does not convert; the unit only labels the answer.
Practice
vmbo a = 6, b = 8. Find c.
c = 10
36 + 64 = 100, √100 = 10.
havo c = 13, a = 5. Find the other leg.
b = 12
√(169 − 25) = √144 = 12.
vwo Can c = 5 when a = 5 and you are looking for a leg?
No
Then b² = 0. The hypotenuse must be strictly longer.
In class
On vmbo you mainly practice 3-4-5 triangles and geometry contexts (ladder, screen diagonal). On havo and vwo the same theorem returns for distances in the plane. Official requirements are in the Dutch math exam programs on Examenblad.nl.
Next step
- How does this work? The explanation and examples are further up this page.
- See it visually Pythagoras simulation: change the values and watch what happens.
- Test yourself Pythagoras practice, with an explanation after every answer.
- Video explanations On the Pythagoras page. YouTube only loads when you press play.
Learning path
- Next: SOHCAHTOA calculator Sides and angles in a right triangle with sine, cosine and tangent.
- Related: Area calculator · Volume calculator
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle: a² + b² = c². c is the hypotenuse.
How do you find the hypotenuse?
c = √(a² + b²). Example: 3 and 4 give √25 = 5.
How do you find a leg?
b = √(c² − a²). c must be longer than a.
Does Pythagoras work for a non-right triangle?
No. Without a right angle, a² + b² = c² does not hold. Check for a right angle first.