Pythagorean Calculator

The 3-4-5 method is a practical application of the Pythagorean theorem to check right angles on construction sites. If two sides of a triangle measure 3 and 4 units, the diagonal must be exactly 5 units for a perfect 90-degree angle. This works with any unit of measurement - meters, centimeters, or even feet.

It is based on the Pythagorean theorem: a2 + b2 = c2. For the 3-4-5 triangle: 32 + 42 = 9 + 16 = 25 = 52. Any triangle with sides in this ratio necessarily has a right angle. This is mathematically proven and has been used in construction practice for thousands of years.

There are two ways: (1) If you know two sides, calculate the third using the Pythagorean theorem: a2 + b2 = c2. (2) If you know one angle and one side, calculate the rest using sine, cosine and tangent. Our calculator does both automatically.

The hypotenuse (c) is the longest side of a right triangle - it lies opposite the right angle (90 degrees). The two shorter sides are called catheti (a and b). The hypotenuse is always longer than either cathetus individually.

With careful measurement, accuracy is within 1-2 mm, which is more than sufficient for most construction work. The main error sources are tape sag over long distances and imprecise marking of measurement points. With a taut tape measure and precise markings, the method is surprisingly accurate.

Essentially just a tape measure (at least 5 meters long), chalk or a pencil for marking, and optionally a chalk line for long distances. No expensive special tools are required - that is one of the great advantages of the 3-4-5 method over laser angle finders.

Laser angle finders are faster but significantly more expensive (starting at around EUR 100). A large carpenter's square works well for short distances up to about 1 meter. The 3-4-5 method is free, works at any scale, and requires no batteries or calibration.

The ancient Greek mathematician Pythagoras proved that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The 3-4-5 triangle is the simplest integer example and therefore ideal for practical application on the construction site.

Laser angle finders are worthwhile for repeated measurements at the same point, such as in series production or drywall with many walls. For one-off checks on the construction site, the 3-4-5 method is simpler, cheaper, and independent of batteries or technology.

The four most common mistakes: (1) Not measuring exactly from the corner point, (2) tape sag over long distances, (3) imprecise marking of measurement points, and (4) confusing which side is A and which is B. Always check the starting point first and keep the tape measure taut.