Category: Geometry

Knowing the value of π is crucial to accurately measuring flow through a round pipe. The formula for determining the volumetric flow through a pipe is Q = A * v. Here Q is volumetric flow, A is the cross-sectional area of the pipe, and v is the average velocity of the flow. We need to know π because the formula for calculating the cross-sectional area of a round pipe is π * r². So, to calculate the volumetric flowrate Q in a round pipe, we multiply (π * r²) * v, where v is the average flow velocity. This applies to mass flow too, since we can derive mass flow from volumetric flow * density.

The history of π goes back more than 4,500 years. Around the year 2000 B.C., the Babylonians and the Egyptians had attempted to determine the value of π. Since these early times, there have been many efforts to calculate π. Since that time, the value of π has attracted the minds of many great thinkers over the centuries, including Gauss, Newton, and the incredibly prolific Swiss mathematician Leonard Euler. Around 1779 Johann Lambert proved that π is irrational. In 1882, F. Lindemann proved that π is transcendental, thereby proving the impossibility of squaring the circle.

Clearly it is important to know the value of π, and trying to understand and determine this value has attracted the attention of some of the greatest minds over the past 4,500 years. But how did the need to know π come about in the first place? Knowing the value of π is the key to knowing the area of a circle. This area is given by the formula A = π * r².

The early Babylonians and the Egyptians found it impossible to measure the length of 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 the prevalent one for many centuries. Eventually, with the invention of calculus and infinite series, the focus turned 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. On “Pi Day” (March 14, 2024), a California storage compan called Solidigm announced that it had calculated π to 105 trillion digits. The calculations took 75 days to complete and used up  1 million gigabytes of data. If you types this number out in 10 point type on a piece of paper, it would be 2.3 billion miles long. This broke the previous record set by Google Cloud in 2022, which had calculated π to 100 trillion digits.

The Role of π in Calculating the Area of a Circle

The following figure shows why there is a need to use π in calculating the area of a circle:

Figure 11-2 The value r² as a unit for measuring the area of a circle

The area of the square with four sides each equaling r is obviously less than the area of the circle whose radius is r. More significantly, the formula requires that some number of squares fit into the circle with radius r. Yet just as you can’t fit a square peg into a round hole, seemingly no definite number of squares will fit into that circular area. The only number that works in this formula is π, an irrational nonrepeating decimal, according to traditional mathematics.

The diagram above shows the difficulty in using a square as a unit of measure for the circle. It was believed that the value of π as an irrational number can never be determined because there is no rational number of squares that will fill the area of a round circle. Also, it shows the difficulty in arriving at an exact value for π, since it is seen as the ratio of the circle to the diameter, and  people couldn’t find a way to measure the length of the circle exactly.

Because π was defined as a ratio of Circumference/Diameter, people tried to find a ratio that equaled π. Examples include 22/7 and 355/113. Leibniz and others even created convergent series that closely approximated π. The problem is that none of these ratios or convergent series equaled  π exactly. So it was assumed to be an irrational nonrepeating decimal number. Calculating π to 105 trillion digits is an admirable feat of technology, but it gets us no closer to understanding π than does the number 3.1416.

If only we could measure the length of the circumference of a circle.

Calculating a Rational Value for the Area of a Circle

I would like to go back to “The Rope Experiment” to explain more completely how this gives us a rational value for the area of a circle.

The Rope Experiment. Start with a rope shaped as a circle with a radius of 2. Now cut the rope and lay it flat, forming a straight line. In algebra, the length of a line is determined by the value of the endpoint minus the starting point, where L is the length of the line:

L = (endpoint – starting point).

In other words, the length of the line is equal to the distance between the endpoint and the starting point of the line. If a line starts at the point (0, 0) in a coordinate system and ends at 4 along the x axis, the line is 4 inches long, assuming we are measuring in inches.

Flatten the Curve. The circumference of a circle with a radius of 2 inches as a rope when laid flat will be 12.5664 inches long. This is algebraically how we measure this distance. There is no need to bring π into this measurement. It is only when we take the flattened rope and shape it back into a circle that its length is then described as 2 * π * r. Yet the rope doesn’t change length when it is made back into a circle.

Every measurement requires a unit of measurement and a degree of precision. We can make a measurement as precise as we like, but we can’t keep shifting the degree of precision so it goes to infinity (as occurs in Zeno’s paradox). Once we specify the unit of measurement and the degree of precision, then the measurement we make is definite – it is not an approximation. If we want a more precise measurement, we can use a higher degree of precision. In the rope example, if the rope is 12.5664 inches long, the unit of measurement is inches and the degree of precision is 1/10000. We can measure π to any degree of precision we want, but once we specify our degree of precision, the measurement is done – there’s no need to think of it as shorthand for an infinitely long decimal.

Now that we have a rational length for the circumference of the circle with radius 2, which is 12.5664 inches, we can derive a rational value for the area of a circle with radius 2. We know that C=2 * π * r. Since r = 2,  this gives us C = 4 * π. This allows us to solve for π, since π = C/4. We have found by flattening the circle that C, the circumference of the circle, is 12.5664 inches. Since 12.5664/4 = 3.1416, the value of pi = 3.1416.

Now that we know the value of π, we can apply this to the formula π * r^2, giving us the area of a circle with radius 2. Since r^2 = 4, the area of this circle is 4 * 3.1416 = 12.5664 square inches.

The definition of π is that π is the ratio of the circumference of a circle to its diameter. The reason pi is irrational in traditional math is because we are comparing a curved or circular shape to a straight line. When we “flatten the curve” and discover its true length as a straight line, the irrationality of the number disappears. This means we can find a rational value for pi to any desired degree of precision, and this also allows us to find a rational value for the area of a circle.