Pythagorean Theorem Calculator
Please provide any 2 values below to solve the Pythagorean equation: a2 + b2 = c2.
Pythagorean Theorem
The Pythagorean theorem, sometimes referred to as Pythagoras’ rule, connects the three sides of a right‑angled triangle. In a triangle where one angle measures 90°, the theorem states that the area of the square drawn on the longest side—the hypotenuse—is exactly the sum of the areas of the squares built on the other two sides.
In other words, given that the longest side c = the hypotenuse, and a and b = the other sides of the triangle:
a2 + b2 = c2
Called the Pythagorean equation after the ancient Greek thinker Pythagoras, this relationship is useful whenever two sides of a right triangle are known, allowing the length of the third side to be calculated. Referring to the diagram above, if
a = 3 and b = 4
the length of c can be determined as:
c = √a2 + b2 = √32+42 = √25 = 5
It follows that the length of a and b can also be determined if the lengths of the other two sides are known using the following relationships:
a = √c2 - b2
b = √c2 - a2
The cosine rule is a broader form of the Pythagorean theorem that lets you determine any side of a triangle when the lengths and opposite angles of the other two sides are known. If the angle between those sides happens to be a right angle, the cosine rule collapses back into the standard Pythagorean equation.
There are a multitude of proofs for the Pythagorean theorem, possibly even the greatest number of any mathematical theorem.
Algebraic proof:
In the figure above, there are two orientations of copies of right triangles used to form a smaller and larger square, labeled i and ii, that depict two algebraic proofs of the Pythagorean theorem.
In the first one, i, the four copies of the same triangle are arranged around a square with sides c. This results in the formation of a larger square with sides of length b + a, and area of (b + a)2. The sum of the area of these four triangles and the smaller square must equal the area of the larger square such that:
| (b + a)2 = c2 + 4 |
| = c2 + 2ab |
which yields:
| c2 = | (b + a)2 - 2ab |
| = | b2 + 2ab + a2 - 2ab |
| = | a2 + b2 |
which is the Pythagorean equation.
In the second orientation shown in the figure, ii, the four copies of the same triangle are arranged such that they form an enclosed square with sides of length b - a, and area (b - a)2. The four triangles with area
| ab |
| 2 |
| (b - a)2 + 2ab | ||||||
| = | b2 - 2ab + a2 + 2ab | ||||||
| = | a2 + b2 |
Since the larger square has sides c and area c2, the above can be rewritten as:
c2 = a2 + b2
which is again, the Pythagorean equation.
There are numerous other proofs ranging from algebraic and geometric proofs to proofs using differentials, but the above are two of the simplest versions.