Find the hypotenuse c, or missing leg a or b, using the Pythagorean theorem (a² + b² = c²). Shows angles, area, perimeter, Pythagorean triple check, and surd form. Supports US and UK units.
The Pythagorean theorem is arguably the most famous equation in mathematics: a² + b² = c². It states that in any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides (the legs). Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), this theorem has been known and used for thousands of years across multiple civilisations.
Our free Pythagorean theorem calculator lets you find the missing side instantly. Choose whether to find c (the hypotenuse), leg a, or leg b — enter the two known sides — and the calculator returns the missing side, both acute angles, area, perimeter, a Pythagorean triple check, and the surd (simplified radical) form of the result.
The core formula in three forms:
Example: if a = 5 feet and b = 12 feet, then c = √(25 + 144) = √169 = 13 feet exactly. This is the famous 5-12-13 Pythagorean triple.
The most intuitive geometric proof of the Pythagorean theorem uses squares drawn on each side of a right triangle. If you construct a square on leg a, a square on leg b, and a square on hypotenuse c, the area of the c-square exactly equals the combined area of the a-square and b-square. This is visible in the classic proof attributed to Euclid (Book I, Proposition 47 of the Elements) and in countless textbook diagrams.
Another elegant proof: arrange four identical right triangles in a square. The outer square has side (a+b) and area (a+b)². The inner tilted square has side c and area c². The four triangles each have area ½ab. So: (a+b)² = c² + 4×(½ab) → a² + 2ab + b² = c² + 2ab → a² + b² = c². Q.E.D.
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². These produce right triangles with whole-number side lengths, making them extremely useful in construction and measurement.
| Triple | a | b | c | Check |
|---|---|---|---|---|
| Simplest | 3 | 4 | 5 | 9+16=25 ✓ |
| Classic | 5 | 12 | 13 | 25+144=169 ✓ |
| Common | 8 | 15 | 17 | 64+225=289 ✓ |
| Common | 7 | 24 | 25 | 49+576=625 ✓ |
| Common | 20 | 21 | 29 | 400+441=841 ✓ |
| Multiple | 6 | 8 | 10 | 36+64=100 ✓ |
| Multiple | 9 | 12 | 15 | 81+144=225 ✓ |
Multiples of Pythagorean triples are also triples. Every multiple of (3,4,5) — such as (6,8,10), (9,12,15), (12,16,20) — satisfies the theorem. The general formula for generating primitive triples is: a = m²−n², b = 2mn, c = m²+n² for any positive integers m > n.
The Pythagorean theorem extends to three dimensions. The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) in 3D space is:
d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)
This is simply the Pythagorean theorem applied twice. In 2D, the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is the direct application of a²+b²=c² where the horizontal and vertical distances are the legs and the straight-line distance is the hypotenuse.
The most practical application of the Pythagorean theorem is checking that a building corner is exactly 90°. Builders use the 3-4-5 method: measure 3 feet (or metres) along one wall, 4 feet along the other, and the diagonal between those two points should be exactly 5 feet. If it is not, the corner is not square. This technique is used daily by carpenters, bricklayers, and construction workers in both the USA and UK. British Standards (BS 8204) for concrete and screed floors require accurate squareness, and the same principle applies.
Calculating rafter length requires the Pythagorean theorem. If a roof has a horizontal run of 12 feet and a vertical rise of 5 feet, the rafter length = √(12² + 5²) = √169 = 13 feet exactly — another 5-12-13 triple. UK roofing calculations use the same approach in metres. Knowing the exact rafter length before cutting timber saves material waste and ensures a correctly proportioned roof structure.
Ships and aircraft use the theorem to find straight-line distance. A plane flying 200 miles east then 150 miles north is √(200² + 150²) = √(40000+22500) = √62500 = 250 miles from the start. Modern GPS systems compute distances using the same principle at the foundation of their algorithms (extended to 3D and adjusted for Earth's curvature for long distances).
In England, Wales, and Northern Ireland, the Pythagorean theorem is a mandatory topic in GCSE Mathematics at both Foundation and Higher tiers (AQA, Edexcel, OCR, WJEC). Students are expected to find missing sides, identify right triangles, and apply the theorem in real-world contexts. Scotland covers it in National 4 and National 5 qualifications.
In the United States, the Pythagorean theorem appears in the Common Core State Standards for Mathematics at Grade 8 (8.G.B.7 and 8.G.B.8). It also features prominently in high school geometry and is tested on the SAT and ACT mathematics sections.
When the result of √(a²+b²) is not a perfect integer, it can be expressed in surd (simplified radical) form. For example, √50 = √(25×2) = 5√2. Surd form gives the exact answer without rounding error, which is important in proof work, exact calculations, and UK A-level mathematics. Our calculator shows both the decimal approximation and notes whether the result is a perfect integer.
| Scenario | US Units | UK/Metric |
|---|---|---|
| Check a 12×16 ft room corner | Diagonal = √(144+256) = 20 ft | 12×16 ft = 3.66×4.88 m; diag = 6.10 m |
| Rafter: 8 ft run, 6 ft rise | Rafter = √(64+36) = 10 ft | 2.44 m run, 1.83 m rise; rafter = 3.05 m |
| Plot: 30 m × 40 m | Diagonal = 164 ft (50 m) | Diagonal = 50 m |
The Pythagorean theorem states that in any right triangle, a² + b² = c², where a and b are the two legs and c is the hypotenuse (the longest side, opposite the right angle). It allows you to find any one side if the other two are known.
Hypotenuse c = √(a² + b²). For example, if the legs are 6 and 8, c = √(36+64) = √100 = 10. If the result is not a perfect square, use a calculator for the decimal: legs 3 and 5 give c = √34 ≈ 5.831.
Pythagorean triples are sets of three whole numbers (a, b, c) satisfying a²+b²=c². The simplest is (3,4,5). Others include (5,12,13), (8,15,17), (7,24,25), and (20,21,29). Any multiple of a triple is also a triple.
Square all three sides and check if the sum of the two smaller squares equals the square of the largest. For sides 9, 40, 41: 81 + 1600 = 1681 = 41². Yes, it is a right triangle. If a²+b² does not equal c², the triangle is not right-angled.
Builders use the 3-4-5 rule to check that corners are square (90°). Mark 3 units on one wall, 4 units on the adjacent wall, and the diagonal between those points should be exactly 5 units. This works in any units — feet, metres, or any consistent measure.
In three dimensions, the distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂) is d = √((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²). The diagonal of a box with dimensions l×w×h is d = √(l²+w²+h²), which applies the theorem twice.
In UK GCSE Mathematics, students learn to apply a²+b²=c² to find missing sides of right triangles, calculate diagonal lengths in rectangles and 3D shapes, and use the theorem in practical problems. It is a mandatory topic at both Foundation and Higher tier.
If the hypotenuse is not a whole number, express it in surd form by finding the largest perfect square factor. For example, √48 = √(16×3) = 4√3. This gives the exact value without rounding. For UK A-level exams, surd form is often required for full marks.
Results are based on mathematical formulas and your inputs. For construction or engineering applications, always verify with professional measurements and consult qualified professionals.