Pythagorean Theorem מחשבון

איך להשתמש במחשבון זה

1. הזן Side a (leg)
2. הזן Side b (leg, optional)
3. לחץ על כפתור חשב
4. קרא את התוצאה המוצגת מתחת למחשבון

The Pythagorean Theorem Explained

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides: a² + b² = c². This theorem, known for over 4,000 years, is one of the most fundamental and widely applied results in all of mathematics.

To find the hypotenuse: c = √(a² + b²). To find a leg when hypotenuse is known: a = √(c² - b²). Our calculator uses the two legs to find the hypotenuse. Example: a right triangle with legs 3 and 4 has hypotenuse √(9+16) = √25 = 5. This 3-4-5 is the most famous Pythagorean triple — whole-number solutions to a² + b² = c².

Other common Pythagorean triples: 5-12-13, 8-15-17, 7-24-25. Any multiple of a triple is also a triple: 6-8-10 (×2 of 3-4-5). These triples were known to Babylonian mathematicians as far back as 1800 BCE, predating Pythagoras himself by 1,200 years.

Proofs and Generalizations

The Pythagorean theorem has over 370 documented proofs — more than any other theorem. Famous proofs include: the dissection proof (rearranging triangle copies to show areas), the algebraic proof using similar triangles, and President James Garfield's 1876 trapezoid proof. Even Einstein devised his own proof at age 12.

The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² - 2ab·cos(C), where C is the angle between sides a and b. When C = 90°, cos(C) = 0 and we recover a² + b² = c². The law of cosines enables solving any triangle given sufficient information.

In higher dimensions, the distance formula extends Pythagoras: the distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂) is √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²). In n-dimensional space: d = √(Σ(xᵢ-yᵢ)²). This is the Euclidean distance, fundamental to machine learning (k-nearest neighbors, k-means clustering).

Practical Applications

The Pythagorean theorem is used in construction and navigation daily. Carpenters use the 3-4-5 rule to ensure corners are square: measure 3 feet along one wall, 4 feet along the adjacent wall, and the diagonal should be exactly 5 feet if the corner is a right angle. GPS navigation calculates distances using the distance formula derived from Pythagoras.

In engineering, the theorem calculates diagonal braces, cable lengths, ramp angles, and structural support sizes. Pilots use it to calculate direct distances: if a plane flies 300 km north then 400 km east, the straight-line distance back to start is √(300² + 400²) = √250,000 = 500 km.

In physics, the Pythagorean theorem applies to vector decomposition, calculating resultant forces and velocities. When a projectile has horizontal velocity vx and vertical velocity vy, its speed is √(vx² + vy²). In special relativity, spacetime distance combines spatial and temporal components using a similar formula (with a minus sign).

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