How it works
Why the Pythagoras calculadora exists
A pythagoras calculadora saves you from squaring, adding and rooting by hand when you already know two sides of a right triangle. GCSE and A-level papers love ladder-against-a-wall, coordinate-distance and “is this triangle right-angled?” questions — all are Pythagoras in disguise.
Three worked examples
Find the hypotenuse — legs 6 cm and 8 cm
c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm — the classic 6-8-10 triple.
Find a shorter side — hypotenuse 13 m, one leg 5 m
a = √(13² − 5²) = √(169 − 25) = √144 = 12 m.
Check if a triangle is right — sides 9, 12, 15
Test 9² + 12² = 81 + 144 = 225 = 15². Yes — it is a right triangle (a 3-4-5 scaling).
Common exam traps
- Hypotenuse is never the shortest side — always identify the right angle first.
- Pythagoras is only for right triangles — for general triangles use cosine rule.
- Units — square both sides in the same unit before adding.
Works well with
- **Hypotenuse calculadora** — same maths, different layout.
- **Triangle area** — ½ × base × height for right triangles.
- **Triangle perimeter** — add all three sides once you know them.
- **Quadratic formula** — distance formula leads to quadratics in harder problems.
Proving the theorem: three short arguments
Pythagoras is one of the most proved results in the history of mathematics — the book *The Pythagorean Proposition* collects over 370 distinct proofs. Three are worth knowing for exam work and intuition.
The "big square" rearrangement proof
Draw a square of side (a + b). Inside it, arrange four congruent right triangles with legs a and b so their hypotenuses form an inner square of side c. The big square's area can be written two ways: (a + b)² and 4 × (½ab) + c². Equating them gives a² + 2ab + b² = 2ab + c², so a² + b² = c². Pure area bookkeeping — no trigonometry needed.
Euclid's proof (Elements, Book I, Proposition 47)
Euclid builds the squares on each side of the right triangle and shows the big hypotenuse square equals the sum of the two smaller squares by a clever congruent-triangle construction. It's the proof every school textbook references, even if few students work through it in full.
Similar triangles proof
Drop a perpendicular from the right angle onto the hypotenuse. The original triangle splits into two smaller right triangles similar to the whole. Matching side ratios gives a² = c·m and b² = c·n where m + n = c. Adding: a² + b² = c·(m + n) = c². Elegant because it only uses similarity.
From theorem to everyday calculation
Pythagoras isn't just a curiosity — it drives a huge number of real-world calculations. Five scenarios you'll see in UK GCSE papers and in day-to-day life.
The ladder on a wall
A 6-metre ladder rests against a wall with its base 1.5 m from the foot of the wall. How high does the ladder reach? Using c = 6, b = 1.5: a = √(6² − 1.5²) = √(36 − 2.25) = √33.75 ≈ 5.81 m. Safety guidance recommends a 4:1 vertical-to-horizontal ratio, so 1.5 m out for 6 m up is actually close to the limit.
Diagonal of a TV screen
TVs are sold by their diagonal measurement. A 55-inch TV has a diagonal of 55 inches, but its width and height depend on aspect ratio. For 16:9, width = 55 × 16/√(16² + 9²) ≈ 47.9 inches; height ≈ 27.0 inches. Handy when checking if a new TV fits your media unit.
Distance between two cities on a flat map
London is approximately 160 km east and 200 km north of Paris (on a simplified flat projection). The straight-line distance is √(160² + 200²) = √(25,600 + 40,000) = √65,600 ≈ 256 km. For accurate distance on a sphere you'd use the haversine formula, but Pythagoras gives a good first approximation.
Setting out a square foundation
Builders use the 3-4-5 rule to check right angles: measure 3 m along one wall, 4 m along the adjoining wall; if the diagonal measures exactly 5 m, the corner is square. Because 3² + 4² = 9 + 16 = 25 = 5², the rule is just Pythagoras applied backwards.
Ramp length for a disability access
A ramp needs to rise 150 mm over a horizontal distance of 1,800 mm (the UK building regs gradient of 1:12). The ramp's physical length is √(1,800² + 150²) = √(3,240,000 + 22,500) ≈ 1,806 mm. Surprisingly close to the horizontal distance because the rise is small.
Distance, coordinates and 3D Pythagoras
In two dimensions, the distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It's just Pythagoras where the "legs" are the horizontal and vertical separations.
Extended to three dimensions, the distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) is √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). GCSE and A-level often use this to find the space diagonal of a cuboid: for a box of width w, depth d and height h, the diagonal is √(w² + d² + h²). A 30 × 40 × 120 storage box has a space diagonal of √(900 + 1,600 + 14,400) = √16,900 = 130 cm.
Converse of Pythagoras and triangle classification
The converse of the theorem is equally useful: if a² + b² = c² for three positive sides a, b, c, then the triangle must be right-angled at the vertex opposite c. This gives a quick test without needing angles or trigonometry.
If a² + b² > c², the triangle is acute (all angles under 90°). If a² + b² < c², it's obtuse (one angle over 90°). Try it with sides 5, 7, 9: 25 + 49 = 74 vs 81. Because 74 < 81, the triangle is obtuse at the corner opposite 9.
Pythagorean triples you should memorise
Examiners reuse the same handful of integer triples because the arithmetic is clean. Knowing them saves precious exam seconds.
| Triple | Common scalings | Appears in |
|---|---|---|
| 3-4-5 | 6-8-10, 9-12-15, 12-16-20 | Foundation GCSE, construction |
| 5-12-13 | 10-24-26, 15-36-39 | Higher GCSE, ladder/ramp problems |
| 8-15-17 | 16-30-34 | A-level mechanics, coordinate geometry |
| 7-24-25 | 14-48-50 | Advanced GCSE |
| 20-21-29 | — | Enrichment problems |
| 9-40-41 | — | A-level surprise problems |
A short history of the theorem
Although called the "Pythagorean" theorem, the result was known long before the Greek philosopher's lifetime (around 570–495 BCE). A Babylonian clay tablet known as Plimpton 322, dated around 1800 BCE, lists what appear to be Pythagorean triples — more than a thousand years before Pythagoras. Indian texts like the *Baudhayana Sulbasutra* (around 800 BCE) include the statement for altar construction. Chinese mathematicians referred to a specific case (3-4-5) as the *gougu* rule in the *Zhou Bi Suan Jing*.
The Pythagorean school in Samos and later Croton is credited with the first general proof. Proclus, writing centuries later, attributed the theorem to the school even if its individual authorship remains debated.
Variations students meet at A-level and beyond
Pythagoras generalises into several important relatives — worth knowing if maths is your thing.
- Law of cosines — c² = a² + b² − 2ab·cos(C). Reduces to Pythagoras when angle C is 90°.
- Law of sines — a/sin(A) = b/sin(B) = c/sin(C). Pairs with cosines for any triangle.
- Pythagorean identities in trigonometry — sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ. Direct descendants of a² + b² = c² applied to a unit circle.
- Higher-dimensional Pythagoras — in n dimensions, the diagonal of a hypercube with side s is s·√n.
- Pythagorean theorem in metric spaces — the defining equation for Euclidean geometry vs non-Euclidean alternatives.
Common exam paper layouts
UK GCSE papers reuse a handful of question styles. Recognising the pattern often unlocks the method.
"Find the missing side"
The simplest case. A 2-mark or 3-mark question with two sides given and a right-angle symbol on the diagram. Worth full marks only if you show the squaring, addition, square-root and state units.
"Is this triangle right-angled?"
Three sides given; apply the converse. Show a² + b² and c² separately, compare, and write a clear statement.
"Coordinate distance"
Given two points, calculate the distance. Write the distance formula, substitute carefully with brackets around negatives, and evaluate.
"Ladder / ramp / flagpole"
Word problem with a picture. Label the right triangle on your own diagram if the paper's is small, and make the hypotenuse explicit before substituting.
Quick reference card
Keep these identities handy for exam prep or quick checks:
- Hypotenuse: c = √(a² + b²).
- Shorter side: a = √(c² − b²), b = √(c² − a²).
- Distance in 2D: d = √((x₂ − x₁)² + (y₂ − y₁)²).
- Diagonal of a rectangle (w × h): d = √(w² + h²).
- Space diagonal of a cuboid (w × d × h): D = √(w² + d² + h²).
- Converse: a² + b² = c² ⇒ right-angled triangle (at vertex opposite c).
Accuracy and sources
We follow BBC Bitesize and exam-board specifications (AQA, OCR, Edexcel). See editorial policy and corrections. Calculations run in your browser using native JavaScript `Math.sqrt`, which is accurate to the 16th decimal place for the inputs supported here.