Solve for a missing hypotenuse or leg, or check whether three sides form a right triangle with this Pythagorean theorem calculator.
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Pythagorean theorem calculator Use this page to find a missing hypotenuse, solve for a missing leg, or check whether three side lengths form a right triangle.
Side-only right-triangle checks
Solve or verify a side set from the Pythagorean theorem
Keep this workflow focused on side lengths: solve the missing side when two are known, or test whether all three sides satisfy a² + b² = c².
Quick examples
Mode
Right-triangle reminder
The theorem only applies to right triangles. In solve modes, the hypotenuse must be longer than the known leg. In check mode, the calculator treats the largest entered side as the hypotenuse candidate before comparing the squares.
Enter valid triangle sides Provide the two known positive sides of a right triangle. If you are solving for a leg, the hypotenuse must be longer than the known leg.
Pythagorean theorem calculator: find the missing side of a right triangle
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This page also explains the main assumptions behind the pythagorean theorem calculator result, highlights the supporting figures shown by the calculator, and helps the reader use the estimate without overstating what a quick online tool can prove.
The Pythagorean theorem
The theorem applies exclusively to right triangles, which have one 90-degree interior angle. The two shorter sides meeting at the right angle are called legs and are labelled a and b. The longest side, opposite the right angle, is the hypotenuse c. Knowing any two sides allows the third to be calculated exactly using the theorem.
To find a missing leg, rearrange the formula. If the hypotenuse c and leg b are known, leg a equals the square root of (c squared minus b squared). If the two legs are known, the hypotenuse is the square root of their sum of squares. Common integer solutions are called Pythagorean triples; the 3-4-5 right triangle is the simplest example, where 9 + 16 = 25.
c^2 = a^2 + b^2
The square of the hypotenuse equals the sum of the squares of the two legs.
a = sqrt(c^2 - b^2) or b = sqrt(c^2 - a^2)
Rearranged forms for finding a missing leg when the hypotenuse and the other leg are known.
Area = 0.5 x a x b
The area of a right triangle is half the product of its two legs.
Perimeter = a + b + c
The perimeter is the sum of all three sides.
Pythagorean triples and real-world uses
Pythagorean triples are sets of three positive integers that satisfy the theorem exactly: 3-4-5, 5-12-13, 8-15-17, and 7-24-25 are common examples. Builders and surveyors use the 3-4-5 rule to verify that corners are square: measuring 3 units along one wall and 4 units along the adjacent wall, the diagonal should be exactly 5 units if the corner is at 90 degrees.
In navigation, the theorem calculates straight-line distances. A ship that sails 30 km east and 40 km north ends up 50 km from its starting point. In screen technology, display diagonal size is found by applying the theorem to width and height. That is why a right triangle calculator or Pythagorean theorem calculator remains useful far beyond school maths: it solves practical distance, layout, and measurement problems quickly.
How to check whether three sides form a right triangle
A good Pythagorean theorem calculator should not only solve for a missing side. It should also test whether three entered sides already satisfy the theorem. The standard approach is to sort the side lengths, treat the largest as the hypotenuse candidate, square all three sides, and compare the largest square with the sum of the two smaller squares.
For example, the side set 8, 15, and 17 passes the check because 64 + 225 = 289. The side set 2, 3, and 4 fails because 4 + 9 = 13 while 4 squared is 16. That difference shows the side set does not form a right triangle, even though all three numbers are positive and could still make a different kind of triangle.
If x ≤ y ≤ z, test x^2 + y^2 = z^2
Sort the side lengths first, then compare the square of the largest side with the sum of the squares of the other two.
Why the square check is useful in practice
Many users are not starting from a neat textbook problem with one blank side. They already have three measured lengths from a drawing, room corner, roof brace, ladder setup, or screen dimension and need to know whether those measurements behave like a right triangle. The square check answers that question directly instead of forcing the user to rearrange the theorem mentally first.
This is also why common triples matter. If the solved or checked side set is close to 3-4-5, 5-12-13, 8-15-17, or a 45-45-90 ratio, that gives the user a fast confidence check. If it is not close to a familiar pattern, the decimal output is still valid, but the user knows to rely on the exact calculation rather than a mental shortcut.
Worked example: solving a 3-4-5 triangle
If the two legs of a right triangle are 3 and 4, the theorem gives c squared = 3 squared + 4 squared = 9 + 16 = 25. The missing hypotenuse is therefore 5. From there, the perimeter is 3 + 4 + 5 = 12 and the area is 0.5 × 3 × 4 = 6.
This kind of worked example is why the calculator is helpful. It turns the theorem into a quick answer for side length, then also gives the other basic triangle measures without doing the arithmetic separately.
Worked example: checking a measured side set
Suppose you measure a corner and get sides of 6, 8, and 10. Square each side: 36, 64, and 100. Because 36 + 64 = 100, the side set passes the Pythagorean check and represents a right triangle. This is the same 3-4-5 pattern scaled up by a factor of 2.
Now compare that with 6, 8, and 9. Squaring gives 36, 64, and 81. Since 36 + 64 = 100, not 81, that set fails the theorem check. This is why the side-set verification mode matters: it helps separate a true right triangle from a triangle that only looks close at first glance.
Frequently asked questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
Can I use this calculator to find a missing leg rather than the hypotenuse?
Yes. Rearranging the formula: a = √(c² - b²). Enter the hypotenuse and one leg and the calculator will find the other leg. This is useful for carpentry, construction, and any right-triangle geometry problem.
How do I tell if three sides make a right triangle?
Sort the three side lengths from smallest to largest, square each one, and compare the largest square with the sum of the two smaller squares. If they match, the side set forms a right triangle. If they do not match, it does not.
Does the theorem only apply to right-angled triangles?
Yes. The theorem is valid only for right-angled triangles. For other triangles, use the law of cosines: c² = a² + b² - 2ab cos(C), where C is the angle between sides a and b. The Pythagorean theorem is a special case where cos(C) = 0 (90-degree angle).
What is a Pythagorean triple?
A Pythagorean triple is a set of three whole numbers that exactly satisfy a² + b² = c². Common examples include 3-4-5, 5-12-13, and 8-15-17.
Can I use decimals in a Pythagorean theorem calculator?
Yes. The theorem works with decimals, fractions converted to decimals, and any consistent length unit. Whole-number triples are just convenient mental-check examples, not a requirement.
How do I find the area of a right triangle from the sides?
If you know the two legs of a right triangle, the area is half the product of those legs: 0.5 × a × b. If you only know the hypotenuse and one leg, you must first solve for the missing leg before finding the area.
What if the square root is not a whole number?
That is normal. Many valid right triangles do not produce whole-number side lengths. The calculator returns the decimal result, and you can still use it for area, perimeter, and measurement checks.
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