when i divide the circumference of a circle by its diameter i get the value of pi and this also happens when i use different units. why does this happen?
When you divide a length by a length, you get a dimensionless number that should always be the same, regardless of units used, as long as the two lengths have the same dimensions.
Many important laws of physics and engineering can be expressed, and are often discovered, using dimensionless numbers.
The fact that dividing the circumference of a circle by its diameter always results in the value of pi, regardless of the units used, is a fundamental mathematical property of circles.
To understand why, let's start with the definition of pi. Pi (π) is an irrational number that represents the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It is a mathematical constant that has been studied for thousands of years and has numerous applications in various fields of science and mathematics.
The formula for the circumference of a circle is given by C = 2πr, where r is the radius. Dividing both sides of the equation by the diameter (D = 2r), we get C/D = (2πr)/(2r).
Now, notice that the 2's on the right side of the equation cancel out, leaving us with C/D = π. So, no matter what unit of measurement you use for the circumference and diameter, the ratio C/D will always be equal to π.
This holds true because pi is the same mathematical constant for any circle, regardless of its size or units of measurement. It is a fundamental property of circles and a consequence of their geometric nature.
Therefore, whenever you divide the circumference of a circle by its diameter, you will consistently obtain the value of pi, regardless of the units used. This fact demonstrates the universality and mathematical elegance of pi.