Pythagoras Theorem Visualiser
Find any missing side of a right triangle and watch the three squares drawn to scale. Check whether three sides make a right triangle, generate Pythagorean triplets and solve real-life ladder and field problems.
Pythagoras' theorem says that in a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². With legs 6 cm and 8 cm, c = √(36 + 64) = √100 = 10 cm. To find a leg instead, subtract: a = √(c² − b²).
Right triangle
a and b are the two legs (sides that make the right angle); c is the hypotenuse.
Squares on each side
Drawn to scale — the two smaller squares together have the same area as the biggest one.
Step-by-step working
Visual proof by rearrangement
Four copies of your triangle inside a square of side (a + b). Slide them around — the empty space never changes.
Do these sides make a right triangle?
Enter three side lengths in any order.
Working
Triplet generator
Euclid’s formula: for whole numbers m > n > 0, the numbers m² − n², 2mn and m² + n² always form a Pythagorean triplet.
Make one from m and n
Triplet containing a number
The Class 8 method: for any whole number ≥ 3.
Pythagorean triplets
| # | a | b | c | m, n | Type | a² + b² = c² |
|---|
Solution
See why a² + b² = c²
The theorem is about areas: the square on the hypotenuse has exactly the same area as the two other squares put together.
Choose the unknown
Pick the hypotenuse or one of the legs, then type the two sides you know.
Watch the squares
The triangle and its three squares are drawn to scale, with unit grids for small whole numbers.
Check and explore
Test any three sides, list Pythagorean triplets and try the rearrangement proof.
Apply it
Solve ladder, field, walking and pole problems with your own numbers.
The theorem
In a right-angled triangle, c² = a² + b² where c is the hypotenuse — the longest side, opposite the right angle.
To find a leg: a = √(c² − b²)
The converse
If the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.
- longest² = sum of other squares → right
- longest² < sum → acute
- longest² > sum → obtuse
Pythagorean triplets
Whole numbers like (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25). Multiply any triplet by the same number to get another: (6, 8, 10), (9, 12, 15).
For any m > 1: 2m, m² − 1, m² + 1 is a triplet (NCERT Class 8).
Pythagoras Theorem Visualiser — your questions
What is Pythagoras' theorem?
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides, written c² = a² + b². The hypotenuse is the longest side and always lies opposite the right angle. It only works for right-angled triangles, never for acute or obtuse ones.
How do I find the missing side of a right triangle?
If you want the hypotenuse, add the squares and take the square root: c = √(a² + b²). If you want a leg, subtract instead: a = √(c² − b²). For a triangle with hypotenuse 13 and one leg 5, the other leg is √(169 − 25) = √144 = 12.
What is the converse of Pythagoras' theorem?
If the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled. The comparison also classifies other triangles: if the longest side squared is less than the sum, the triangle is acute; if it is greater, the triangle is obtuse. Sides 6, 8, 10 give 100 = 100, so it is right-angled.
What are Pythagorean triplets?
They are sets of three whole numbers that satisfy a² + b² = c², such as (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25). Multiplying any triplet by the same number gives another one, so (3, 4, 5) also gives (6, 8, 10) and (9, 12, 15).
How do I generate a Pythagorean triplet?
The NCERT Class 8 rule uses any whole number m greater than 1: the three numbers 2m, m² − 1 and m² + 1 always form a triplet. Taking m = 4 gives 8, 15 and 17, and 8² + 15² = 64 + 225 = 289 = 17². The tool lists triplets generated this way.
How is Pythagoras' theorem used in ladder problems?
The wall, the ground and the ladder form a right triangle with the ladder as the hypotenuse. If a 13 m ladder stands with its foot 5 m from the wall, it reaches √(169 − 25) = 12 m up the wall. The same set-up solves pole, kite-string, field diagonal and walking-distance questions.