Everyone knows Pythagorean theorem about right triangle. We often use it to determine a right angle.

a²+b²=c²

you will find out here that sometimes aproximate equation is better:

a²+b²≈c²

If we have two sides with lengths of 1, the diagonal will have length of √2 ≈ 1.4142135623731, Simple, right? We have calculators after all.

The side lengths are defined in any unit, such as millimeters, centimeters, meters, kilometers, Roman miles, or even feet, cubits, or other body part lengths.

However, measuring fractional lengths is inconvenient. Where exactly is the value 1.4142 located between the lines?

In the table 1 you can calculate the value of the hypotenuse c and angle α.

In the table 4 you will find that triple (70,70,99) gives decent results in this case.

In practice, we try to use integer values and for this purpose we use so-called Pythagorean triples. (3,4,5), (5,12,13), (7,24,25) etc. These are integers where the sum of the squares of two of them is equal to the square of the third number.: 3²+4²=5², 5²+12²=13², 7²+24²=25²

In this case, we can, for example, prepare a rope with knots that will determine our own unit of measurement. We can also use ready-made elements with evenly spaced holes, or use a template to drill holes for screw connections.

In the table 2 you can calculate Pythagorean triples.

However, using integer lengths doesn't solve the problem. In the real world, we deal with measurement errors, and we can't absolutely determine the lengths of the triangle sides. It is difficult to match the lines on the ruler, and the hole-bolt connections have unavoidable play. The right angle will be determined with error. We also have no control over the angle α.

In the table 3 you can calculate the value of the hypotenuse c and 90° angle error for a given accuracy of the dimensions.

Therefore, assuming maximum error in determining angles, we can use integer lengths that are not necessarily Pythagorean triples, but that determine a right angle and (if desired) the angle α with sufficient accuracy. This is useful whenever we use a quantized unit of measurement.

In the table 4 you can calculate the value of the hypotenuse c and angles α and 90° for a given accuracy of the dimensions..

On page 5 You can download the program to perform calculations on your local computer.