The Hesapstan Pythagorean theorem calculator is designed to find the third side of a right triangle from the two you already know, using the identity a² + b² = c². Enter the two legs and leave the hypotenuse field empty to get c = √(a² + b²); enter the hypotenuse and one leg instead and it solves for the remaining leg as √(c² − a²). If you fill in all three sides there is nothing left to solve, so the calculator applies the converse of the theorem and tells you whether those three lengths form a right triangle, judged at the decimal precision you actually entered. Results are marked with ≈ when the displayed value has been rounded, the answer carries whatever unit your sides use, and the tool never assumes a right angle on your behalf.
What does this calculator do?
The calculator offers three side fields: the two legs a and b, and the hypotenuse c. Leave the side you want to find empty and fill in the other two. The intermediate steps — the squares and their sum or difference — appear as separate rows, and the side you were looking for is marked as the main result.
- Enter a and b, leave c empty → the hypotenuse c = √(a² + b²) is computed.
- Enter c and one leg → the remaining leg is computed as √(c² − a²).
- Enter all three sides → the calculator checks whether the triangle is right-angled.
- Every side you enter must be positive, and when solving for a leg the hypotenuse must exceed the known leg.
- A result rounded to the displayed number of digits is prefixed with ≈.
What is the Pythagorean theorem?
The Pythagorean theorem states that for any right triangle, the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². It is one of the most widely used results in geometry.
- a = 3, b = 4 → c = √(9 + 16) = √25 = 5 (the 3-4-5 triple).
- a = 5, b = 12 → c = √(25 + 144) = √169 = 13 (the 5-12-13 triple).
- a = 1, b = 1 → c = √2 ≈ 1.4142135624.
Worked example
Suppose the two legs are a = 9 and b = 12. Square each leg first: a² = 81 and b² = 144. Then add the squares: 81 + 144 = 225. Finally take the square root of the sum: c = √225 = 15. The answer is a whole number here because (9, 12, 15) is a Pythagorean triple — it is the (3, 4, 5) triple scaled by three.
- a² = 9² = 81.
- b² = 12² = 144.
- a² + b² = 81 + 144 = 225.
- c = √225 = 15.
Change one leg slightly to b = 12.5 and the sum becomes 81 + 156.25 = 237.25, giving c ≈ 15.4029. The calculator then shows the ≈ marker to signal that the value is rounded rather than exact. Note too that the hypotenuse is always longer than either leg, which is a quick way to sanity-check any result.
When is the theorem used?
The Pythagorean theorem is used in geometry, physics, engineering, and many everyday measurements. Whenever the horizontal and vertical distances between two points are known, the straight-line distance is given by the Pythagorean theorem. Diagonals of rectangles and slope-based distance calculations also rely on the same identity.
The theorem only holds for right triangles. When you ask the calculator to solve for a missing side it assumes the right angle is there; it can only verify that condition when you enter all three sides.
How to use the calculator
- Enter the lengths of the two sides you already know.
- Leave the field for the side you want to find empty.
- Use decimal input if needed — either a comma or a dot is accepted.
- Read the result area: the intermediate squares, then the side you were solving for, shown as the main result.
- Fill in all three sides instead and the calculator reports whether the triangle is right-angled, judged at the precision you entered.
Common mistakes
- Entering a zero or negative side: the calculator flags non-positive inputs and computes nothing.
- Applying the theorem to non-right triangles: the identity only holds for triangles with a 90° angle — use the law of cosines otherwise.
- Typing the hypotenuse into a leg field: the hypotenuse is always the longest side and belongs in its own field.
- Solving for a leg with a hypotenuse shorter than the known leg: no such triangle exists and the calculator refuses it.
- Confusing c with an arithmetic mean: c is the square root of a squared sum, not an average.
Limitations
This calculator works only with the right-triangle identity. It finds the third side from the two you know, and when you supply just two sides it must assume the triangle really is right-angled — there is no third measurement available to verify that. The right-angle check runs only when all three sides are entered, and it respects the number of decimals you typed: if √(a² + b²) rounds to the hypotenuse you entered at that precision, the result is reported as consistent rather than as a plain yes. The tool reports no angles, no area and no perimeter, and it does not test any other triangle-construction condition. Side lengths are bounded in magnitude by 1e15. No official data, API, or external source is used.
Frequently Asked Questions
What is the Pythagorean theorem?
It states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c².
Does the theorem apply to non-right triangles?
No. It only holds for triangles with a 90° angle.
What is the 3-4-5 triple?
It is a right triangle with sides 3, 4, and 5 because 3² + 4² = 9 + 16 = 25 = 5².
Can I enter decimal side lengths?
Yes. For example, a = 2.5 and b = 6 gives exactly c = 6.5. Both a comma and a dot are accepted as the decimal separator.
What happens with a zero or negative side?
The calculator shows a note that sides must be positive and computes nothing.
Can I find a leg when the hypotenuse is known?
Yes. Enter the hypotenuse and one leg, leave the other leg field empty, and the missing leg is computed as √(c² − a²).
Does this calculator verify the right-angle condition?
It does when you enter all three sides. The check respects the precision you typed: for a = 3 and b = 5 the exact hypotenuse is √34 ≈ 5.8309518948, so entering c = 5.83 is a correct rounding at two decimals and the result is reported as consistent at that precision, while c = 5.82 or c = 5.84 is rejected. If you enter only two sides and let it solve for the third, the right angle is assumed, because there is no third measurement to check it against.
What does the ≈ symbol before a result mean?
It indicates that the displayed value has been rounded to the number of digits shown. For a = b = 1, for instance, c = √2 cannot be written exactly and appears as ≈ 1.4142135624.