Pythagorean Theorem Calculator
Enter any two values of a right triangle: two sides, or one side and one acute angle. You get the other sides, both angles, the area, the perimeter, the altitude to the hypotenuse and the inradius, and a note when the sides form a Pythagorean triple such as 3, 4, 5. Everything runs in your browser.
3, 4, 5 is a primitive Pythagorean triple: all three sides are whole numbers with no common factor.
Pythagorean triples with a hypotenuse up to 100
Bold triples are primitive, which means the three numbers have no common factor. The others are multiples of them.
The Pythagorean theorem says that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a squared plus b squared equals c squared. With two sides known, the third follows, and the angles come from the sine, cosine and tangent. With one side and one acute angle, the other sides come from trigonometry. All sides use one unit of your choice, and angles are in degrees. Everything is worked out in your browser and nothing is uploaded.
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Math & MoneyAbout the Pythagorean Theorem Calculator
The Pythagorean theorem says that in a right triangle the square of the hypotenuse c equals the sum of the squares of the legs a and b, so c squared = a squared + b squared. With two sides known, the third follows: the hypotenuse is the square root of a squared plus b squared, and a missing leg is the square root of c squared minus the other leg squared. The two acute angles come from the inverse tangent, and they always add up to 90 degrees. With one side and one acute angle, the other sides come from the sine, cosine and tangent: a leg opposite angle A is the hypotenuse times sin A, and the other leg is the hypotenuse times cos A. Two angles alone cannot fix the size of a triangle, because every similar triangle has the same angles, so the tool asks for a side as well. The hypotenuse must be longer than either leg, and an acute angle must be between 0 and 90 degrees. The area is half of a times b, the altitude to the hypotenuse is a times b divided by c, and the radius of the inscribed circle is (a + b - c) divided by 2. When all three sides are whole numbers, the tool says it is a Pythagorean triple, and a primitive one when the numbers share no common factor, such as 5, 12, 13. A list of the triples up to a hypotenuse of 100 is included. Use any unit, as long as it is the same for every side. Nothing is uploaded.
How do I find the hypotenuse of a right triangle?▾
Square both legs, add them, and take the square root. For legs of 3 and 4, 9 + 16 = 25, and the square root of 25 is 5. Enter the two legs in the tool and it gives the hypotenuse straight away.
How do I find a missing leg?▾
Square the hypotenuse, subtract the square of the known leg, and take the square root. With a hypotenuse of 13 and a leg of 5, 169 - 25 = 144, and the square root is 12. The hypotenuse must be longer than the leg.
Can I solve a right triangle from an angle and a side?▾
Yes. Enter one acute angle and any one side. For a hypotenuse of 10 and an angle of 30 degrees, the opposite leg is 10 times sin 30, which is 5, and the other leg is about 8.66. You cannot solve it from two angles, because they only give the shape, not the size.
What is a Pythagorean triple?▾
Three whole numbers a, b and c with a squared + b squared = c squared, such as 3, 4, 5 or 5, 12, 13. A primitive triple has no common factor, and every other triple is a multiple of one, so 6, 8, 10 is double 3, 4, 5.
Does the theorem work for any triangle?▾
No, only for right triangles, which have one 90 degree angle. For other triangles use the law of cosines, which reduces to the Pythagorean theorem when the angle is 90 degrees. Nothing you enter is uploaded.