Free maths tool · Level: Ages 13 to 15
Pythagoras theorem
a² + b² = c² is about squares, and they are rarely drawn. Here they are. Drag the triangle and check that it holds.
In a right-angled triangle the square on the hypotenuse has the same area as the two squares on the other sides put together: a² + b² = c². So the hypotenuse is c = √(a² + b²), and it is always the longest side, opposite the right angle.
It is about areas, not lengths
a² is not just a number times itself, it is the area of a real square with side a. The theorem says two of those squares exactly fill the third.
Drag the two points on the sides. The shape changes, the numbers change, but the blue area plus the amber area stays exactly equal to the red one. That is the whole theorem.
Which side is the hypotenuse
The hypotenuse is always opposite the right angle and always the longest. The most common mistake is putting a or b where c belongs.
Quick check: your c should be bigger than a and bigger than b, but smaller than a + b. Outside that range, something has gone wrong.
The whole-number triples that keep appearing
Some combinations give tidy whole numbers: 3-4-5, 5-12-13, 8-15-17. Exam questions are often built on them to keep the arithmetic clean.
Set a to 3 and b to 4 and the tool calls it out. Spotting a triple gives you the third side with no calculation. Note that multiples count too, so 6-8-10 works just as well.
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Frequently asked questions
- What does Pythagoras theorem say?
- In a right-angled triangle, a² + b² = c², where c is the hypotenuse. In areas: the squares on the two shorter sides exactly fill the square on the hypotenuse.
- How do you calculate the hypotenuse?
- Square both shorter sides, add them, and take the square root: c = √(a² + b²). With a = 3 and b = 4 that gives √25 = 5.
- How do I find a shorter side if I know the hypotenuse?
- You subtract instead of adding: a = √(c² - b²). The square on the hypotenuse is the biggest, so the smaller one comes off it.
- Does it work for any triangle?
- No, only right-angled ones. Without a right angle the equality breaks down and you need the cosine rule instead.
More interactive maths tools
- Discriminant and parabola →Slide a, b and c and see D decide whether the parabola cuts, touches or misses the axis.
- Unit circle →Drag the angle and see sin, cos and tan as lengths, with the exact values.
- Derivative and tangent →Drag the point and watch the tangent tilt, with the secant closing in on it.