The most famous theorem in geometry: in any right triangle, a² + b² = c². Find any one side given the other two.
Hypotenuse c
5
a² + b² = c²
कुछ सामग्री अब भी अंग्रेज़ी में दिखती है।
The Pythagorean theorem (a² + b² = c² for any right triangle) is arguably the single most-applied formula in all of mathematics - used in construction, navigation, computer graphics, physics, surveying, carpentry and everyday measurement. Indian mathematics records the relationship in the Baudhayana Sulba Sutra (c. 800 BCE), centuries before Pythagoras's Greek school formalised it. Whether you're computing the diagonal of a TV screen, the shortest path across a rectangular garden, the length of a roof rafter, or the distance between GPS coordinates over short distances, the Pythagorean theorem is the underlying math.
This calculator works in both directions: given two legs (the perpendicular sides), find the hypotenuse (longest side). Given the hypotenuse and one leg, find the other leg. Class 7-10 mathematics in India relies heavily on this formula - Class 7 introduces it, Class 9 proves it geometrically, Class 10 boards regularly test applications. JEE/NEET aspirants encounter it in coordinate geometry (distance formula is just Pythagoras applied to coordinate differences), 3D geometry, vectors, and physics problems involving force components.
Pick which side is unknown - hypotenuse or one of the legs.
Enter the other two sides.
Read the missing side.
If you know the two legs, hypotenuse = √(a² + b²). If you know the hypotenuse and one leg, the other leg = √(c² − leg²). The hypotenuse is always the longest side.
a² + b² = c²
Statement of theorem: in any right-angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides: a² + b² = c², where c is the hypotenuse (always opposite the right angle, always the longest side) and a, b are the two legs (the perpendicular sides forming the right angle).
Finding the hypotenuse: c = √(a² + b²). Example: legs 3 and 4, hypotenuse = √(9+16) = √25 = 5. Finding a leg: a = √(c² - b²). Example: hypotenuse 13, one leg 5, other leg = √(169-25) = √144 = 12. The 3-4-5 and 5-12-13 are the two most-famous Pythagorean triples - integer solutions where all three sides are whole numbers.
Converse and validation: if a² + b² = c², the triangle is right-angled at the vertex opposite c. If a² + b² > c², the triangle is acute. If a² + b² < c², it's obtuse. This converse is useful for verifying if measured sides form a right triangle (commonly used by carpenters to check 90° corners using the 3-4-5 rule). 3D extension: the space diagonal of a rectangular box with dimensions l, w, h is d = √(l² + w² + h²) - Pythagoras applied twice.
Width:height = 16:9 in some scaling. Width² + height² = 32². With width = 16k, height = 9k: (16k)² + (9k)² = 256k² + 81k² = 337k² = 1024. k = √(1024/337) ≈ 1.74. Width ≈ 27.9 inches; height ≈ 15.7 inches.
Mark 3 feet along one side, 4 feet along the other. If diagonal between marks = exactly 5 feet, corner is square (90°). Used worldwide by builders to verify foundation corners without protractors.
By Pythagoras: wall_reach² = 5² - 3² = 25 - 9 = 16. Wall reach = 4m. Used in safety calculations - IS 3696 (Indian Standard for ladder safety) recommends slope angle 75° (1:4 base-to-height ratio).
No - only right triangles. For others, use the law of cosines: c² = a² + b² − 2ab·cos(C).
Pythagoras (~570 BC) gets the credit, but the relationship was known much earlier in Babylonian, Indian (Sulba Sutras) and Chinese (Zhou Bi Suan Jing) mathematics.
Calculating diagonal of a TV / monitor, mounting brackets, finding the shortest distance between two points on a flat plane, GPS distance over short distances (with corrections).
Not a triangle, or your hypotenuse is wrong. The hypotenuse must equal exactly √(a²+b²). If c is bigger, the sides don't form a closed triangle.
Sets of three integers satisfying the theorem: (3,4,5), (5,12,13), (8,15,17). Used in carpentry to mark perfect 90° angles without instruments.
The diagonal of a rectangular box is d = √(l² + w² + h²) - Pythagoras applied twice.
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