Pythagorean Theorem Calculator

3 squared plus 4 squared, and two ways to get it wrong

3 squared is 9 and 4 squared is 16, so 3 squared plus 4 squared is 25, and its square root, 5, is the hypotenuse. The order is what trips people up: square each leg, add the squares, then take one square root at the end. Two shortcuts look plausible and both give 7 instead of 5. Adding the legs first gives (3 + 4) squared, which is 49, because (a + b) squared expands to a squared plus 2ab plus b squared and carries an extra 2 times 3 times 4, or 24. Taking square roots term by term gives the square root of 9 plus the square root of 16, which is just 3 plus 4 again. 7 is the distance along both legs, and the diagonal is always shorter than that, which makes a quick check on any answer.

Pythagorean triples: right triangles with whole-number sides

A Pythagorean triple is a set of three whole numbers that fit a squared plus b squared equals c squared exactly. The table lists every primitive triple (one with no common factor) whose hypotenuse is 65 or less. Multiplying all three numbers in a row by the same amount gives another triple, so 3, 4, 5 also produces 6, 8, 10 and 9, 12, 15, and these scaled versions are the ones that turn up most in exam questions and on building sites.

65 appears twice because it is the smallest number that is the hypotenuse of two different primitive triples, 16, 63, 65 and 33, 56, 65.

Hypotenuse for common leg lengths

Most pairs of whole-number legs do not give a whole-number hypotenuse, and the answer is an irrational number that the calculator rounds to four decimal places. When both legs are equal the triangle has two 45 degree angles and the hypotenuse is always the leg times the square root of 2, about 1.4142, which is also the diagonal of a square with that side.

Checking a right angle with the 3-4-5 rule

The theorem also works in reverse. If the three sides of a triangle satisfy a squared plus b squared equals c squared, the angle between a and b is exactly 90 degrees. Builders, tilers and gardeners use this to set out a square corner without a large set square: measure 3 units along one side and 4 along the other, and adjust until the diagonal between the two marks is exactly 5. The same check scales to 0.6, 0.8 and 1.0 metres for a small job or 6, 8 and 10 feet for a slab or a deck. Setting out a floor this way before working out how many tiles it needs with the tile calculator stops the cut row along the far wall from tapering.

If the diagonal comes out long, the corner is wider than 90 degrees (obtuse), because a squared plus b squared is then less than c squared. If it comes out short, the corner is too tight (acute). For sides of 3 and 4, a diagonal of 5.1 means the corner is obtuse, since 25 is less than 26.01, and 4.9 means it is acute, since 25 is more than 24.01.

How high a ladder reaches

A ladder leaning on a wall is a right triangle: the ladder is the hypotenuse, the wall is one leg and the ground between the wall and the foot is the other. Ladder safety guidance in both the UK and the US uses the 1 in 4 rule, which puts the foot about a quarter of the ladder's working length out from the wall, giving an angle of roughly 75 degrees. The height reached is then the square root of the length squared minus the quarter-length squared, which works out at about 97 percent of the ladder's length whatever the unit.

Lengths are in any one unit, metres or feet. Height reached is where the top of the ladder touches the wall, not a safe standing height, and the guidance also asks for the top to extend past a landing point you step off at. Check the label on your own ladder.

TV and monitor screen sizes

Screens are sold by their diagonal, which is the hypotenuse of the triangle formed by the width and the height. A 16:9 screen is a 16 by 9 right triangle scaled up, and its diagonal is the square root of 256 plus 81, which is the square root of 337, about 18.3576. Dividing the advertised size by that figure and multiplying by 16 and by 9 gives the width and the height. The ratio calculator handles other aspect ratios.

These are screen dimensions. The bezel and stand add to the footprint, so check the manufacturer's spec sheet before buying for a tight alcove.

Distance between two points

The distance formula from coordinate geometry is the Pythagorean theorem in disguise. The horizontal gap between two points is one leg, the vertical gap is the other, and the straight line between them is the hypotenuse, so the distance is the square root of the x difference squared plus the y difference squared. From (1, 2) to (7, 10) the gaps are 6 and 8, and the distance is the square root of 36 plus 64, which is 10. Enter the two gaps as the legs above to get any other distance. In three dimensions the same idea adds a third squared gap under the square root, so the diagonal of a 3 by 4 by 12 box is 13.