Pythagorean Theorem Calculator
Please provide any 2 values below to solve the Pythagorean equation: a² + b² = c².
Modify the values and click the Calculate button to use
Pythagorean Theorem
The Pythagorean Theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
Example: If a = 3 and b = 4, then:
c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5
To find a:
a = √(c² - b²)
To find b:
b = √(c² - a²)
The Pythagorean Theorem is a special case of the Law of Cosines, which applies to any triangle.
Algebraic proof:
The Pythagorean Theorem can be proven algebraically using geometric arrangements. The following diagrams demonstrate two key arrangements that lead to the proof.
Diagram (i):
Diagram (ii):
In Diagram (i), we have a large square with side length (b + a). Inside it, four identical right triangles are arranged, leaving a central square with side c. The area of the large square is (b + a)².
In Diagram (ii), we have a square with side c. Inside it, the same four right triangles are arranged, leaving a central square with side (b - a). The area of the large square is c².
Since both diagrams contain the same four right triangles, the remaining areas must be equal. This gives us:
This completes the algebraic proof of the Pythagorean Theorem, showing that the sum of the squares of the two shorter sides equals the square of the hypotenuse in any right triangle.