Pythagorean Theorem Calculator

Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’ theorem, is a fundamental relation between the three sides of a right triangle.
In a right triangle (where one angle is 90°), the theorem states that the square of the hypotenuse (the longest side, c) is equal to the sum of the squares of the other two sides (a and b):

Formula:

a² + b² = c²

This equation helps determine the length of one side if the other two are known. For example, if:

a = 3  
b = 4  

Then the hypotenuse c is:

c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5

Solving for Other Sides:

• a = √(c² – b²)
• b = √(c² – a²)

The Law of Cosines generalizes this theorem and can calculate any triangle’s side given two sides and the included angle. If the angle is 90°, it simplifies to the Pythagorean Theorem.

Algebraic Proofs:

The Pythagorean theorem has many proofs. Below are two algebraic proofs based on rearranging four right triangles to form squares.

Proof i: Square with side (a + b)

Four triangles form a large square with area: (a + b)².

The remaining inner square has side c and area c².
The total area is also made of 4 triangles, each with area (ab/2).

(a + b)² = c² + 4 × (ab/2)  
         = c² + 2ab  
Therefore:  
c² = (a + b)² - 2ab  
   = a² + 2ab + b² - 2ab  
   = a² + b²

Proof ii: Square with side (b – a)

Triangles form a square with sides c and area c². The inner square has side (b – a) and area (b – a)². Each triangle still has area (ab/2).

c² = (b - a)² + 4 × (ab/2)  
   = (b - a)² + 2ab  
   = b² - 2ab + a² + 2ab  
   = a² + b²

These are just two of the many elegant proofs of the Pythagorean Theorem. Others involve geometry, algebra, and even calculus.

Related Calculators:
Surface Area Calculator, Circle Calculator

External Resources:
Pythagorean Theorem Calculator on Calculator.net