The Baudhayana–Pythagoras Theorem | Mathematics Grade 8 CBSE

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The six ideas of the theorem

From the parts of a right triangle to its ancient Indian origin and the whole-number triples that satisfy it — tap each term to see how it fits.

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The three big ideas

A right triangle has one 90° angle. The two sides forming it are the legs; the side opposite is the hypotenuse — always the longest side.
The rule: the square drawn on the hypotenuse has the same area as the two squares on the legs combined → a² + b² = c².
Baudhayana's Sulbasutra — an ancient Indian text (composed centuries before the Greek mathematician Pythagoras) — stated this rule in the language of altar-building. "Sulba" means a cord, because the altars were laid out with stretched ropes.
In his own words (paraphrased): the rope along the diagonal of a rectangle makes a square whose area equals the squares made by the two sides together — exactly a² + b² = c².

A Pythagorean triple is a set of three whole numbers that fits a² + b² = c². Classic ones: (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25).
Multiples also work — multiply every number by the same whole number: (3, 4, 5) → (6, 8, 10) → (9, 12, 15), all triples.
Finding the hypotenuse: c = √(a² + b²).
Finding a leg: rearrange to a = √(c² − b²).

Worked example. A ladder 13 m long leans against a wall, its foot 5 m from the base. How high up the wall does it reach?

1. The wall, ground and ladder form a right triangle. The ladder is the hypotenuse (c = 13 m); the distance from the wall is one leg (b = 5 m).

2. Find the other leg: a² = c² − b² = 13² − 5² = 169 − 25 = 144.

3. a = √144 = 12 m.

4. So the ladder reaches 12 m up the wall — this is the (5, 12, 13) triple.

Common mistake: adding the sides instead of their squares — writing 5 + 12 = 13. The theorem uses the squares: 5² + 12² = 13², not 5 + 12.

The converse: if the three sides of a triangle satisfy a² + b² = c² (c the longest), then the triangle must be right-angled, with the right angle opposite the longest side.
This turns the theorem into a test: take any three lengths, square the two shorter, add them — if the total equals the square of the longest, you have a right angle.
Example: sides 9, 12, 15 → 9² + 12² = 81 + 144 = 225 = 15² → right-angled.
Builders use the "3-4-5 method": mark 3 units along one edge, 4 along the other; if the diagonal between them is 5 units, the corner is a true right angle.

Where you'll meet it

The theorem at work

Squaring a building's corners

Masons laying a foundation use the 3-4-5 rope trick: measure 3 m and 4 m along two edges; when the diagonal reads exactly 5 m, the corner is a perfect right angle. The same idea Baudhayana used for altars builds straight walls today.

Shortest path across a field

Walking diagonally across a rectangular field 30 m by 40 m beats going round the edges: the diagonal is √(30² + 40²) = 50 m, versus 70 m along two sides. The theorem measures the straight-line shortcut.

Ramps, ladders & cables

How long a ramp is needed to rise 1 m over a 2.4 m run? √(1² + 2.4²) = √6.76 = 2.6 m. Engineers size ramps, ladders and bridge cables with exactly this right-triangle calculation.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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