The Baudhayana Pythagoras Theorem
Long before Pythagoras, the Indian mathematician Baudhayana stated the relationship between the sides of a right-angled triangle. This Class 8 chapter explores that theorem: in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, and shows how to use it to find unknown lengths and to check right angles.
Learning objectives
- State the Baudhayana–Pythagoras theorem.
- Identify the hypotenuse and the legs of a right triangle.
- Find an unknown side using the theorem.
- Recognise and use Pythagorean triples.
Key concepts
The theorem
In a right-angled triangle, the side opposite the right angle is the hypotenuse and is the longest side. The theorem states that the square on the hypotenuse equals the sum of the squares on the other two sides: hypotenuse² = base² + height². Baudhayana described this rule in the Sulba Sutras centuries before Pythagoras.
Finding an unknown side
If two sides are known, the third follows from the theorem. With legs 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5. If the hypotenuse and one leg are known, subtract: the other leg is √(hypotenuse² − leg²).
Pythagorean triples
Three whole numbers that fit a² + b² = c² form a Pythagorean triple, such as (3, 4, 5), (5, 12, 13) and (8, 15, 17). Any multiple of a triple is also a triple, so (6, 8, 10) works too. Triples make quick right-triangle problems.
Checking for a right angle
The theorem also works in reverse: if the sum of the squares of two sides equals the square of the third, the triangle is right-angled. Builders use the 3-4-5 rule to set out perfect right angles, since 3² + 4² = 5².
Important formulas
Pythagoras theorem
hypotenuse² = base² + height² (c² = a² + b²)
Unknown leg
a = √(c² − b²)
Right-angle test
if a² + b² = c², the triangle is right-angled
Key definitions
- Hypotenuse
- The side opposite the right angle in a right triangle; the longest side.
- Right-angled triangle
- A triangle with one angle equal to 90°.
- Pythagorean triple
- Three whole numbers a, b, c satisfying a² + b² = c².
- Leg
- Either of the two sides that form the right angle.
Solved examples
Q1. Find the hypotenuse of a right triangle with legs 6 cm and 8 cm.
Solution: c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Q2. A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
Solution: Other leg = √(13² − 5²) = √(169 − 25) = √144 = 12.
Q3. Is a triangle with sides 9, 12, 15 right-angled?
Solution: 9² + 12² = 81 + 144 = 225 = 15². Yes, it is right-angled.
Common mistakes to avoid
- Taking a leg as the hypotenuse — the hypotenuse is opposite the right angle and is longest.
- Adding the sides instead of their squares.
- Forgetting to take the square root at the end.
- Subtracting in the wrong order when finding a leg (use c² − leg²).
The Baudhayana Pythagoras Theorem — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The longest side of a right triangle is the:
Practice questions
Short answer
State the Baudhayana–Pythagoras theorem.
In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
Find the hypotenuse for legs 5 and 12.
√(25 + 144) = √169 = 13.
Give one Pythagorean triple.
(3, 4, 5) — since 3² + 4² = 5².
Long answer
A ladder 13 m long leans against a wall with its foot 5 m from the wall. How high does it reach? Show the working.
The ladder is the hypotenuse (13 m) and the distance from the wall is one leg (5 m). The height reached is the other leg: height = √(13² − 5²) = √(169 − 25) = √144 = 12 m. So the ladder reaches 12 m up the wall.
Explain how to test whether a triangle is right-angled, using sides 7, 24, 25.
Identify the longest side as the possible hypotenuse (25). Square the two shorter sides and add: 7² + 24² = 49 + 576 = 625. Compare with the square of the longest side: 25² = 625. Since 7² + 24² = 25², the converse of the theorem tells us the triangle is right-angled.
HOTS (Higher Order Thinking)
Two sides of a right triangle are 9 and 12. Could the third side be 15 or could it be something else?
If 9 and 12 are the legs, the hypotenuse is √(81+144) = 15. But if 12 were the hypotenuse and 9 a leg, the other leg would be √(144−81) = √63 ≈ 7.9, so the answer depends on which side is the hypotenuse.
Why is the hypotenuse always the longest side of a right triangle?
It is opposite the largest angle (the 90° angle), and in any triangle the longest side lies opposite the largest angle.
Quick revision
Revision notes
- Hypotenuse² = base² + height² (right triangles only).
- Unknown leg = √(hypotenuse² − leg²).
- Pythagorean triples: (3,4,5), (5,12,13), (8,15,17) and their multiples.
- Converse: a² + b² = c² means the triangle is right-angled.
Key takeaways
- The hypotenuse is opposite the right angle and is longest.
- Square the sides, never just add them.
- The 3-4-5 rule builds exact right angles.
Frequently asked questions
What is the Baudhayana–Pythagoras theorem?
In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
How do I find a missing side?
Use c² = a² + b²; for a leg, rearrange to leg = √(c² − other leg²).
What is a Pythagorean triple?
Three whole numbers like 3, 4, 5 that satisfy a² + b² = c².