a² + b² = c² for right triangles — sides in any length unit.
Enter the two known sides; the unknown is solved automatically.
c = √(a² + b²) | a = √(c² − b²) | b = √(c² − a²)
For any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. To find the hypotenuse c, take the square root of the sum of the squares of legs a and b. To find a missing leg, rearrange the formula: subtract the square of the known leg from the square of the hypotenuse, then take the square root.
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Result: Hypotenuse c = 5 units
When the two legs measure 3 and 4 units, the hypotenuse is exactly 5 units. This is the famous 3-4-5 Pythagorean triple — one of the simplest whole-number solutions to the theorem. The result tells you the straight-line distance between the two far corners of the right triangle, which is always the longest side.
The Pythagorean Theorem states that in any right triangle (a triangle containing a 90° angle), the square of the length of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
The theorem is named after the ancient Greek mathematician Pythagoras of Samos (c. 570–495 BCE), although evidence of the relationship appears in Babylonian and Indian mathematics centuries earlier.
The Pythagorean Theorem is one of the most widely applied equations in mathematics and everyday life:
A Pythagorean triple is a set of three positive integers (a, b, c) that perfectly satisfy a² + b² = c². Common examples include:
| a | b | c | |---|---|---| | 3 | 4 | 5 | | 5 | 12 | 13 | | 8 | 15 | 17 | | 7 | 24 | 25 |
Multiples of any triple also form a triple (e.g., 6-8-10 is a multiple of 3-4-5).
If the three sides of a triangle satisfy a² + b² = c², the triangle must be a right triangle. This is used in construction (the "3-4-5 rule") to confirm that corners are perfectly square.
For a rectangular box with dimensions l, w, and h, the space diagonal d is:
d = √(l² + w² + h²)
This is simply the Pythagorean Theorem applied twice in sequence.
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². The smallest and most famous example is (3, 4, 5). Multiplying all three numbers by the same integer (e.g., 6, 8, 10) produces another valid triple.
Square all three side lengths. If the sum of the two smaller squares equals the largest square — i.e., a² + b² = c² — the triangle is a right triangle by the converse of the Pythagorean Theorem.
It is used to measure diagonal distances (e.g., screen size, staircase length), verify square corners in construction, calculate shortest paths in navigation, and find vector magnitudes in physics and engineering.
They are the same concept. The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is simply the Pythagorean Theorem applied to a right triangle formed by the horizontal and vertical differences between two coordinate points.
Yes. The 3D distance (space diagonal) is d = √(a² + b² + c²), which applies the 2D theorem twice — first to the base and then incorporating the height.
Yes. Select the leg you want to find (a or b), then enter the hypotenuse and the other leg. The calculator rearranges the formula to a = √(c² − b²) or b = √(c² − a²) automatically.
The calculator is unit-agnostic. Enter any consistent unit — centimetres, metres, inches, feet, etc. — and the result will be in the same unit. Just make sure both inputs use the same unit.
That input combination is geometrically impossible. The calculator will flag an error because the expression under the square root (c² − a²) would be negative, meaning no real solution exists.
No. The Pythagorean Theorem applies **only to right triangles**. For triangles with no 90° angle, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
Absolutely. The calculator accepts any positive real number, including decimals like 2.5 or 7.07. The result is rounded to two decimal places.
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