The Pythagorean Theorem Calculator applies a² + b² = c² to a right triangle, solving for the hypotenuse, a missing leg, or a yes-or-no test of whether three given sides form a right angle. Enter any two of the three sides to find the third, or enter all three to run the converse check.
The relation holds only for right-angled triangles, where c is the side opposite the right angle. The calculator shows the substituted squares and the square root so each step can be checked by hand.
Apply the Pythagorean theorem
The Pythagorean theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs. Written with the usual letters, that is a² + b² = c², where c is the longest side. The theorem is the engine behind distance on a coordinate plane, the diagonal of a rectangle, and the slant height of a cone.
This page owns the pure relation and the converse. The Right Triangle Calculator owns the full trig solution when an angle is also involved. Enter the known sides, choose which variable to solve for if needed, and read the result with the squared terms expanded in the working panel.
Solve for the hypotenuse
When both legs are known, the hypotenuse is the positive square root of the sum of their squares. The length is always greater than either leg. Units stay consistent with the inputs: if a and b are in centimetres, c is in centimetres.
c = √(a² + b²)Exact-radical form is offered when the sum of squares is not a perfect square, alongside the decimal value. For integer legs that form a known triple, the result is an integer. For other legs the calculator reports a rounded decimal and keeps full precision internally.
Solve for a missing leg
When the hypotenuse and one leg are known, rearrange the theorem to isolate the unknown leg. The hypotenuse must be strictly longer than the known leg. If c is less than or equal to a, no real right triangle exists with those two lengths, and the tool reports the impossibility instead of returning an invalid root.
b = √(c² - a²)The same rearrangement works with either leg missing. Choose the field that matches the unknown, enter the other two lengths, and the calculator fills the gap.
Test whether a triangle is right-angled
The converse of the Pythagorean theorem says that if three positive lengths satisfy a² + b² = c² when c is the longest, then the triangle with those sides is right-angled at the vertex between a and b. Enter all three sides to run that check.
The calculator squares each side, compares the sum of the two smaller squares with the largest square, and reports right-angled, acute, or obtuse. A floating-point tolerance covers rounding of measured sides so a near miss is not mislabelled. This test is why the page accepts three inputs even when nothing is missing.
Solve for a triangle with legs 6 and 8
Take legs a = 6 and b = 8. These sides are a scaled 3-4-5 triple, each multiplied by 2, so the hypotenuse is the integer 10 and the arithmetic stays exact through every step.
1. Square each leg. `` 6² = 36 8² = 64 ``
2. Add the squares. `` 36 + 64 = 100 ``
3. Take the square root. `` c = √100 = 10 ``
The area of that right triangle is (1/2) × 6 × 8 = 24, which matches the SSS worked example on the Triangle Calculator page.
Recognise common Pythagorean triples
A Pythagorean triple is three positive integers a, b, c with a² + b² = c². Primitive triples have no common factor. The most common ones students meet are listed below. Any positive integer multiple of a triple is also a right triangle: 6-8-10, 9-12-15, and 10-24-26 are scaled copies.
| Triple | Check |
|---|---|
| 3-4-5 | 9 + 16 = 25 |
| 5-12-13 | 25 + 144 = 169 |
| 8-15-17 | 64 + 225 = 289 |
| 7-24-25 | 49 + 576 = 625 |
| 20-21-29 | 400 + 441 = 841 |
| 9-40-41 | 81 + 1600 = 1681 |
Recognising a triple saves time on exams because the hypotenuse needs no calculator once the pattern is spotted.
Use the theorem in coordinate geometry
Place two points on a plane and the horizontal and vertical gaps between them form the legs of a right triangle. The straight-line distance is the hypotenuse. That identity is why the distance formula looks like the Pythagorean theorem with Δx and Δy in place of a and b.
d = √((x₂ - x₁)² + (y₂ - y₁)²)The Distance Calculator owns the full 2D and 3D treatment. The Slope Calculator owns rise over run for the same two points. This page is the place to see why those formulas work: the grid draws a right triangle, and c is the distance.
Frequently asked questions
What is the Pythagorean theorem?
The Pythagorean theorem states that for a right triangle with legs a and b and hypotenuse c, a² + b² = c². The square on the longest side equals the sum of the squares on the other two.
How do I find the hypotenuse?
Square both legs, add the results, and take the positive square root: c = √(a² + b²). For legs 6 and 8, c = √(36 + 64) = √100 = 10.
How do I find a missing leg?
Rearrange to b = √(c² - a²), where c is the hypotenuse and a is the known leg. The hypotenuse must be longer than the known leg or no real solution exists.
How can I tell if three sides form a right triangle?
Order the sides so c is longest, then check whether a² + b² equals c². If it does, the triangle is right-angled. If a² + b² is greater, the triangle is acute. If smaller, it is obtuse.
What are the most common Pythagorean triples?
The triples students meet first are 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and 9-40-41. Multiples such as 6-8-10 are equally valid right triangles.
Does the theorem work for non-right triangles?
No. The equality a² + b² = c² holds only when the angle between a and b is 90 degrees. For other triangles use the law of cosines, which reduces to the Pythagorean theorem when the included angle is a right angle.
How is the theorem related to the distance formula?
The distance between two points is the hypotenuse of the right triangle formed by their horizontal and vertical separations. Expanding that hypotenuse produces √((x₂-x₁)² + (y₂-y₁)²).
Can the calculator solve for either variable?
Yes. Enter any two of a, b and c to find the third, or enter all three to test the converse. Solve-for-any-variable is built into the engine relationships.
Why must c be the longest side?
In a right triangle the hypotenuse faces the largest angle, so it is strictly longer than either leg. Labelling a leg as c breaks the formula and the converse test.
Summary
The Pythagorean Theorem Calculator applies a² + b² = c² to find a hypotenuse or a missing leg, and uses the converse to test whether three sides form a right angle. Legs 6 and 8 give hypotenuse 10. Common integer triples such as 3-4-5 and 5-12-13 are listed for quick recognition. The same relation underpins the coordinate distance formula.