Pythagorean Theorem Calculator

A Pythagorean theorem calculator earns its keep in two directions. Forwards, it takes any two sides of a right triangle and returns the third through a² + b² = c². Backwards, it takes three measured sides and tells you whether the corner between the two shorter ones really is 90 degrees. Most tools stop at the first job. This one does both, draws the triangle to scale with the square on each side, and when the answer to the second question is no, says how far off you are and in which direction.

Two sides in, the third side out

The two legs, a and b, meet at the right angle. The hypotenuse, c, sits opposite it and is always the longest side. With both legs known, square them, add, and take the square root: legs of 7 and 9 give 49 + 81 = 130, and √130 is 11.40 to two places. With the hypotenuse and one leg known, the sum turns into a subtraction: a hypotenuse of 10 and a leg of 6 give 100 - 36 = 64, so the missing leg is 8. That subtraction is why the calculator refuses a hypotenuse shorter than or equal to the leg you typed. The difference would be zero or negative, and no real length squares to a negative number, which is the arithmetic way of saying the hypotenuse must be the longest side.

Where the squared answer is a whole number but not a perfect square, the page also prints the exact radical. Legs of 3 and 6 give c² = 45, and √45 simplifies to 3√5 because 45 is 9 times 5. A decimal such as 6.7082 is rounded; 3√5 is not, which matters when the answer feeds a later step of a homework problem or a proof. Two equal legs of 1 give the diagonal of a unit square, √2, about 1.4142.

Three sides in: the converse, graded rather than yes or no

The converse of the Pythagorean theorem says that if the two shorter sides squared add up to the longest side squared, the angle opposite the longest side is a right angle. Euclid proved the theorem as Proposition 47 of Book I of the Elements and the converse as Proposition 48, around 300 BC. It is the converse, not the theorem, that a builder uses, because on site the angle is the unknown and the lengths are what the tape measures.

The same comparison also sorts every other triangle. If the two shorter sides squared add to more than the longest squared, every angle is under 90 degrees and the triangle is acute. If they add to less, the largest angle is over 90 degrees and the triangle is obtuse. Sides of 8, 15 and 17 pass, since 64 + 225 = 289 = 17². Sides of 8, 15 and 16 fail, because 289 is more than 256, so the triangle is acute. Enter 3, 4 and 6 and the page reports an obtuse triangle whose largest angle is 117.28 degrees and whose longest side is a full unit, or 20 percent, longer than the 5 a right angle needs. Enter 3, 4 and 4.9 and it reports an acute triangle, largest angle 87.64 degrees, 2 percent short. If three lengths cannot close at all, because the two shorter ones add up to no more than the longest, it says so instead of drawing something impossible.

Squaring a foundation with 3 4 5

The 3 4 5 rule is the converse in work boots. From the corner, measure 3 units along one string line and 4 along the other, mark both, and adjust until the diagonal between the marks reads exactly 5. Any multiple works, and larger multiples are better: ConBros lists 6-8-10 for small decks and patios, 9-12-15 for garages and room additions, 12-16-20 for house foundations and 15-20-25 for large foundations, with the diagonals agreeing to within 1/8 inch for most residential work. The reason bigger is better shows up in the converse. A 1/8 inch miss on a 3-4-5 laid out in feet (36, 48 and 60.125 inches) is a corner of 90.25 degrees. The same 1/8 inch on 12-16-20 (144, 192 and 240.125 inches) is 90.06 degrees, four times closer to square for the same tape error.

For the whole rectangle rather than one corner, the diagonal is the check. A 24 by 36 foot foundation has a diagonal of √(576 + 1,296) = √1,872, about 43.27 feet or roughly 43 feet 3 3/16 inches, and when both diagonals measure the same the rectangle is square. Type the length and width as the two legs above to get that figure for any footprint, then switch the unit to feet.

Ladder reach and the 4 to 1 rule

A ladder leaning on a wall is a right triangle with the ladder as the hypotenuse. The United States construction standard, OSHA 29 CFR 1926.1053(b)(5)(i), asks for the foot of a non-self-supporting ladder to sit about one quarter of the working length out from the top support, which works out at roughly 75.5 degrees. The theorem supplies the reach. A ladder with 20 feet of working length stands 5 feet out, and the top support is √(400 - 25) = √375, about 19.36 feet up, not the full 20. The loss is always the same fraction, because the reach is the working length times √15 ÷ 4, or about 96.8 percent of it. The same standard wants the side rails to extend at least 3 feet above an upper landing, so a ladder chosen for access needs that allowance on top of the height it has to reach. The classic textbook version runs the other way: a 13 metre ladder with its foot 5 metres out reaches 12 metres, the 5, 12, 13 triple.

Everything else one right triangle gives you

Once all three sides are known, the rest follows. The area of a right triangle is half the product of the legs, because the legs are a base and its perpendicular height. The two acute angles come from arctan(a ÷ b) and its complement, so 3, 4, 5 has angles of 36.87 and 53.13 degrees. The height dropped from the right angle to the hypotenuse is ab ÷ c, 2.4 for the 3, 4, 5 triangle, and it splits the hypotenuse into pieces of a² ÷ c and b² ÷ c, which are 1.8 and 3.2. The inradius is (a + b - c) ÷ 2 and the circumradius is exactly half the hypotenuse, because the hypotenuse is a diameter of the circle through all three corners. The readout beside the diagram lists each of these, and the working panel shows the squares being added or subtracted line by line.

Pythagorean triples, and why the table has sixteen rows

A Pythagorean triple is three whole numbers that satisfy a² + b² = c². A primitive triple has no common factor, so 3, 4, 5 is primitive and 9, 12, 15 is the same triangle scaled by 3. Euclid's formula produces every primitive triple: pick whole numbers m greater than n, coprime and not both odd, and the sides are m² - n², 2mn and m² + n². There are exactly 16 primitive triples with a hypotenuse of 100 or less, and the reference table above lists all of them with their m and n. Every one shows the same fingerprints: one leg divisible by 3, one leg divisible by 4, one side divisible by 5, so the product of the three sides is always a multiple of 60. The hypotenuse is always odd, the area is a whole number divisible by 6 and the inradius is a whole number. The shape column is there because a layout seldom wants the smallest triple; it wants one whose proportions match the job, and 20, 21, 29 or 65, 72, 97 are the ones to reach for when the corner sits near 45 degrees.

Much older than Pythagoras

The name is a credit, not a history. A Babylonian clay tablet known as Plimpton 322, written around 1800 BC and now held at Columbia University, carries 15 rows that read as the sides of right triangles, including the triple 12709, 13500, 18541. The Baudhayana Shulba Sutra, dated between the 8th and 5th centuries BC, states the rule in India, and in China it is the Gougu theorem. Euclid gave the oldest surviving axiomatic proof. Proofs have multiplied since: Elisha Loomis collected 370 of them in The Pythagorean Proposition, and James Garfield published one in the New-England Journal of Education in 1876, while he was a congressman from Ohio, by working out the area of a trapezoid built from three right triangles in two different ways.

Where this page stops

The theorem holds only when one angle is exactly 90 degrees. For any other triangle the general rule is the law of cosines, c² = a² + b² - 2ab cos C, which collapses back to a² + b² = c² when C is 90 degrees because cos 90° is zero. The straight line distance between two points on a grid is the same theorem applied to coordinates, and it has its own tool in the distance calculator. Rise over run, grade and pitch belong to the slope calculator, and the area of shapes other than a right triangle is covered by the area calculator.