Most people are familiar with the formula for measuring the area of a circle: π * r2, where “r” stands for the radius of the circle. Meanwhile, there’s no escaping the need to measure the area of a round pipe when calculating flow.
The formula for measuring volumetric flow is Q = A * v, where “Q” is the volume of flow that passes a specific point in a unit of time. “A” is the cross-section of the inside of the pipe and “v” is the average flow velocity. The cross-section is calculated as Area = π * r2.
The circumference of a circle is determined by 2 * π * r. It is used find the value of π, which is defined as the ratio of the circumference of a circle to its diameter.
Why we need pi
Early Babylonians and Egyptians found it impossible to measure the circumference of a circle as a round area. Instead, they tried approximating its length with polygons and other straight-line shapes that closely resembled the distance around the circle. This approach remained prevalent for centuries, but the value of π has attracted the minds of great thinkers including Carl Friedrich Gauss, Isaac Newton and Leonard Euler. Eventually, the invention of calculus and infinite series turned the focus to finding a mathematical equivalent to the circumference of the circle. These efforts proved fruitless as well.
Once the computer age arrived in the 1960s, the focus of those seeking a value for π turned to calculating it to as many decimals as possible in hopes of finding a set of repeating digits. In 2024 on Pi Day (March 14), a California-based data storage company, Solidigm, announced it had calculated π to 105 trillion digits. The calculations took 75 days to complete and used up a million gigabytes of data. If you type this number out in 10-point type on a piece of paper, it would be 2.3 billion miles long.
The Rope Experiment
The Rope Experiment provides a way to measure the length of the circumference of a circle that yields a rational value.
For example, a circle with a radius (r) of 2 inches and a diameter (d) of 4 inches gives the following value for the circumference of the circle: C = 2 * π * 2. Multiplying this out, it is: C = 4 * π
Given that π is approximately 3.1416, this gives a value of 12.5664 inches for the circumference of the circle. However, it can only be an approximation since we are using an approximate value for π.
Now, take a rope and form it into a circle with a radius of 2 inches—easily be done by measuring in a straight line from the center of the circle to its edge formed by the rope. Lay the rope that forms the circle flat and it becomes a straight line of length C. The straight line has measurable, rational length (you can measure it with a ruler or counting units). There is no need to use π provide the length of a straight line. So, the circumference is a rational number.
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Measure the rope with a yardstick or a tape measure. You will find that the length of the rope is right around 12.5 inches, depending on required precision. Using the approximate value of π as 3.1614 in the formula 2 * π * r, the circumference is 12.5664 inches. Using the formula for the circumference in the equation 2 * p * r, we get an irrational value that can only be approximated because it includes a value (π) with indefinitely many digits that never repeat.
Use a tape measure to get a value for the length of the rope that is just over 12.5 inches. We can carry this out to any desired level of precision, but it depends on our unit of measurement and the precision of our measuring tool. Whether we use a fraction or a decimal value, the measured number we get is a rational number and it does not contain a never-ending string of decimal values.
Length doesn’t change
A rope doesn’t change its length when you shape it into a circle. The length of the rope remains constant regardless of its shape. The ratio of a circle's circumference to its diameter is defined by the length of the circular boundary measured in rational units. The result is a rational value, specific to the precision of measurement, not a transcendental abstraction. The figure below shows the rope measurement.
Given that we have a measurable, rational length for the rope when it is laid flat, this is the length of the rope. And this is also the length of the rope when it is shaped into a circle. The problem is not with our measurement of the rope—the problem is with our formula for calculating the circumference of a circle.
