Pythagorean Theorem Calculator

The formula and the three sides

The Pythagorean theorem links the three sides of a right triangle: the squares of the two shorter sides add up to the square of the longest side.

a^2 + b^2 = c^2

Side c is the hypotenuse, the side across from the 90 degree corner, and it is always the longest of the three. Sides a and b are the legs, the two sides that meet at the right angle. NASA's Glenn Research Center states the theorem the same way and notes that the square on the hypotenuse has the same area as the two leg squares combined.

The theorem works in one direction only: it applies to right triangles. MIT's Blossoms geometry notes give the area proof: four copies of the triangle sit inside a square of side a + b, and the uncovered area is c^2.

That proof is worth reading once, because it explains why the formula adds squares rather than sides. A triangle with a 91 degree corner does not satisfy it, and using the formula there returns a side that is too long. The last section covers the general version for those triangles.

Finding the hypotenuse

When the two legs are known, add their squares and take the square root:

c = square root of (a^2 + b^2)

A stair stringer is the board that carries the steps. If a staircase rises 7.5 ft over a run of 10 ft, the stringer is the hypotenuse: 7.5^2 + 10^2 = 56.25 + 100 = 156.25, and the square root of 156.25 is 12.5 ft. Buy a 13 ft board and cut it to length.

Squaring before adding is the part that catches people. Adding the sides first gives 17.5, which is a full 5 ft longer than the true diagonal. The square root of a sum is not the sum of the square roots, so 7.5 + 10 is not the answer.

The same arithmetic covers most household jobs. Each row below uses only the formula above.

In every row the unknown is the longest side or a leg, never both. That is the signal for which version of the formula to use: two legs known means add, hypotenuse and one leg known means subtract.

Finding a missing leg

When the hypotenuse and one leg are known, subtract instead of adding:

a = square root of (c^2 - b^2)

An 8 ft ladder stands 2 ft from a wall. The ladder is the hypotenuse and the ground gap is one leg, so the height on the wall is the other leg: 8^2 - 2^2 = 64 - 4 = 60, and the square root of 60 is 7.746 ft. The top of the ladder sits about 7 ft 9 in up the wall.

The subtraction only works if the hypotenuse is the longest side. If c is shorter than b, the square under the root goes negative and there is no real triangle. A calculator that returns a number there is hiding an input error, which is why this page reports "no real triangle" instead.

Area, perimeter and the altitude

Once all three sides are known, the rest of the triangle follows without trigonometry. For a right triangle the legs are the base and the height, so the area is half their product:

area = a x b / 2

A 3-4-5 triangle has area 3 x 4 / 2 = 6 square units and perimeter 3 + 4 + 5 = 12. The altitude drawn from the right angle to the hypotenuse is a x b / c, which is 12 / 5 = 2.4. The circle that fits inside the triangle has radius (a + b - c) / 2 = (3 + 4 - 5) / 2 = 1.

Those four numbers, area, perimeter, altitude and inradius, are what most carpentry and geometry problems ask for. The side length is only the first step.

Whole-number triples are worth memorizing because they skip the square root entirely. The six below are the ones that show up in framing, layout and test questions.

Each row satisfies a^2 + b^2 = c^2 exactly. A 20-21-29 triangle, for instance, gives 400 + 441 = 841, and the square root of 841 is 29. The angles are not round numbers even when the sides are.

Angles from side lengths

A right triangle has one angle fixed at 90 degrees, so the other two add up to 90. Their sizes come straight from the side ratios:

sin A = a / c, cos A = b / c, tan A = a / b

In the 3-4-5 triangle, angle A is the inverse sine of 3/5, which is 36.87 degrees, and angle B is the inverse sine of 4/5, which is 53.13 degrees. The two add to 90, as they must.

The ratios are scale free: a 6-8-10 triangle has the same angles as a 3-4-5 triangle because every side doubled. That is why roof pitches and ramps can be described by rise and run alone, with no lengths attached.

Right, acute or obtuse: the converse

The theorem runs backwards too. Take the longest side as c and compare a^2 + b^2 with c^2. If the two sides are equal, the triangle is right. If the sum is larger, the angle across from c is acute. If the sum is smaller, that angle is obtuse.

The comparison only works with the longest side in the c slot. Put a shorter side there and every triangle looks obtuse. A 6-8-9 triangle is acute because 36 + 64 = 100 is greater than 81, while 6-8-11 is obtuse because 100 is less than 121.

Pythagoras in three dimensions and on screens

The theorem extends to three dimensions by applying it twice. The diagonal of a box is the square root of the sum of all three squared sides:

d = square root of (x^2 + y^2 + z^2)

A box 3 ft by 4 ft by 12 ft has a diagonal of 13 ft, because 9 + 16 + 144 = 169. The same arithmetic sizes a television: a 55 inch set with a 16:9 screen has a diagonal of 55, and the width and height follow from the ratio. The ratio's own diagonal is the square root of 16^2 + 9^2, which is 18.36, so the width is 55 x 16 / 18.36 = 47.9 in and the height is 55 x 9 / 18.36 = 27.0 in.

Check it: 47.94^2 + 26.96^2 = 2,298.2 + 726.8 = 3,025, and the square root of 3,025 is 55. The screen fits a 48 by 27 inch opening, which is why a 55 inch set rarely fits a cabinet built for a 50 inch one.

Where the theorem stops applying

The formula is exact for right triangles and wrong for everything else. For a triangle with sides a and b meeting at angle C, the general rule is the law of cosines:

c^2 = a^2 + b^2 - 2 x a x b x cos C

Two sides of 6 and 8 with a 60 degree angle between them give c^2 = 36 + 64 - 48 = 52, so c = 7.21. The Pythagorean answer would be 10, which is off by almost 3.

Two more limits matter in practice. The theorem measures straight-line distance on a flat surface, so it does not give the shortest path on a sphere or a curved roof. And it assumes the measurements are exact: a 1/8 inch error on each leg of a 3-4-5 triangle moves the hypotenuse from 5 to 5.175 inches, which is fine for a deck and not fine for a machine part.

Units have to match before any of this works. The table below converts the common ones; convert first, then square.

Squaring a number in feet and a number in inches gives a meaningless result. Pick one unit for all three sides before you start.