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Pythagorean Theorem: Formula, Proof, Examples and Uses
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The Pythagorean theorem says that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². Here c is the hypotenuse (the longest side, opposite the right angle) and a and b are the two shorter sides. If you know any two sides of a right triangle, the theorem gives you the third.
Put simply: draw a square on each side of a right triangle, and the two small squares together have exactly the same area as the big square on the longest side.
The Pythagorean theorem formula
| You want | Formula |
|---|---|
| Hypotenuse c | c = √(a² + b²) |
| Side a | a = √(c² − b²) |
| Side b | b = √(c² − a²) |
The theorem only works for right-angled triangles. For other triangles, use the cosine rule, c² = a² + b² − 2ab cos C, which becomes Pythagoras when C = 90° because cos 90° = 0.
Solved examples
Example 1: finding the hypotenuse. A right triangle has legs of 6 cm and 8 cm. c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Example 2: finding a shorter side. A 13 m ladder leans against a wall with its foot 5 m from the wall. How high up the wall does it reach? h = √(13² − 5²) = √(169 − 25) = √144 = 12 m.
Example 3: checking for a right angle. Is a triangle with sides 7, 24 and 25 right-angled? 7² + 24² = 49 + 576 = 625 = 25². Yes, it is. This “converse” of the theorem is how you test whether a corner is truly square.
Example 4: diagonal of a rectangle. A TV screen is 120 cm wide and 68 cm tall. Its diagonal is √(120² + 68²) = √(14,400 + 4,624) = √19,024 ≈ 137.9 cm, about 54 inches (137.9 ÷ 2.54 ≈ 54.3), which is how screen sizes are quoted.
Pythagorean triples
A Pythagorean triple is a set of three whole numbers that fit a² + b² = c². Memorising a few saves time in exams.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Any multiple of a triple is also a triple: 6, 8, 10 and 9, 12, 15 both come from 3, 4, 5.
A simple proof
Take four copies of a right triangle with legs a and b and hypotenuse c. Arrange them inside a large square of side (a + b) so that their hypotenuses form a tilted square of side c in the middle.
- Area of the large square = (a + b)² = a² + 2ab + b².
- The same area also equals the four triangles plus the middle square: 4 × (½ab) + c² = 2ab + c².
- Setting them equal: a² + 2ab + b² = 2ab + c², so a² + b² = c².
What is the Pythagorean theorem used for?
- Construction and carpentry: builders use the 3-4-5 rule to mark a perfect right angle. Measure 3 units along one wall and 4 along the other; if the diagonal is exactly 5, the corner is square.
- Surveying and civil engineering: finding slope lengths, distances across obstacles and the length of a ramp or roof rafter.
- Navigation and maps: the straight-line distance between two points is √(Δx² + Δy²), which is Pythagoras on a grid. GPS and the distance formula in coordinate geometry rely on it.
- Physics and engineering: combining perpendicular vectors, such as finding the resultant of two forces at right angles or the magnitude of a velocity from its components.
- Electronics: in an AC circuit, impedance Z = √(R² + X²) combines resistance and reactance, which are at 90° to each other.
- Computer graphics and games: calculating distances between points for collision detection.
Who discovered it?
The theorem is named after the Greek mathematician Pythagoras (about 570 to 495 BCE), but the relationship was known much earlier. Babylonian clay tablets list Pythagorean triples, and the Indian Baudhayana Sulba Sutra (roughly 800 to 500 BCE) states the rule for the diagonal of a rectangle. Pythagoras or his followers are usually credited with the first general proof.
Common mistakes
- Using it on a triangle that is not right-angled.
- Adding the squares when finding a shorter side; you must subtract from c².
- Forgetting to take the square root at the end.
- Mixing units, such as metres and centimetres, in the same calculation.
Frequently asked questions
What is the Pythagorean theorem in simple words?
In a right triangle, the longest side squared equals the sum of the other two sides squared: a² + b² = c².
What is the formula of the Pythagorean theorem?
The formula is a² + b² = c², where c is the hypotenuse. Rearranged, c = √(a² + b²).
Does the Pythagorean theorem work for all triangles?
No. It works only for right-angled triangles. For other triangles use the cosine rule.
What is the 3-4-5 rule?
It is the smallest Pythagorean triple. Because 3² + 4² = 5², a triangle with sides 3, 4 and 5 always has a right angle, which builders use to set out square corners.
References
- NCERT Class 10 Mathematics, Chapter 6 “Triangles”.
- National Council of Teachers of Mathematics (NCTM)
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