It can be measured by comparing the shape to squares of a fixed size." Sure, it can. Wikipedia defines area as "the quantity that expresses the extent of a region on the plane or on a curved surface. But measuring round areas in squares?! What would you say if in reverse we were measuring square areas by the number of round coins covering them? How is that not as good as the opposite? Introducing circular units of areas It makes perfect sense measuring rectilinear areas in square units – they are a good fit for each other. C’mon! We invented the wheel, electricity, engines, airplanes, telephone, radio, TV, computers, spaceships, artificial intelligence and more, but we are still measuring round areas in squares! Think about it for a minute: for thousands of years we have been measuring round areas by the number of squares with a side=1 covering that round area. It is the presumption that an area must be measured in squares (square inches, square meters, square miles, etc.). However, all these methods have one funny thing in common. There is also a multitude of its proofs using modern methods rooted in trigonometry and calculus. Many famous ancient mathematicians from Archimedes to Eudoxus of Cnidus to Hippocrates of Chios have proven this formula. We all know from middle school’s elementary geometry that area of a circle of radius r is A = πr 2 (or A = π d 2/4 using diameter d since r = d/2 ). Why do we measure area of a circle in squares? Overall, this square/cube-centric language insidiously conditions our minds into thinking only in terms of squares and cubes regarding the units of area and volume measurement. To me this is like calling a round wheel “a rolling square”. Moreover, the very word “area” itself is often replaced by square footage, square mileage, or square something thus further implying that areas and squares are synonymous. So why do we need this suggestive verbal association of the area/volume measurement units to squares and cubes? All natural numbers N have their corresponding root operations, which we call the Nth root. The same semantical problem persists with the reverse terms square root and cube root. They are just abbreviations for multiplication operations and have nothing to do with geometrical shapes and exist independently from geometry and area/volume calculations. Wait a minute! Exponents b 2 and b 3 are mathematical operations from the Algebra Department. Similarly, the expression b 3 = b ⋅ b ⋅ b is called "the cube of b" or " b cubed", because the volume of a cube with side-length b is b 3. “The expression b 2 = b ⋅ b is called "the square of b" or " b squared", because the area of a square with side-length b is b 2. Here is how Wikipedia explains this exponentiation phenomenon: However, besides 2 and 3, we treat all other exponents N p equally by saying "N to the power of p". Have you ever noticed that we call a number to the power of 2 as the number squared? Similarly, we call a number to the power of 3 as the number cubed. Let’s start with some questionable and overreaching semantics in algebra. (Definition: A misnomer is a wrong or inaccurate use of a name or term.) Wanna know how? Just sit down, relax, open your mind and fasten your seat belt. In this post, I dare to debunk the notion of π being an inherent component of the omnipresent circle area and sphere volumes formulas:īelieve it or not, we can significantly simplify these formulas by getting rid of π altogether.
![circles in rectangle optimization w radius of 2 circles in rectangle optimization w radius of 2](https://i.stack.imgur.com/i3hfM.png)
While it is impossible to overstate the significance of π in its fundamental and legendary role of being a constant representing the circumference/diameter ratio, its usage for area and volume measurements is not as much of a “settled science” as you may think. Therefore, we can derive a circle's circumference C from its diameter d using the following famous formula:
![circles in rectangle optimization w radius of 2 circles in rectangle optimization w radius of 2](https://www.geogebra.org/resource/DzJK7q3z/yC3pCLdcF6NEsqN8/material-DzJK7q3z.png)
This date in the format MM/DD is 3/14 which corresponds to the first three digits of the π value 3.14.Īs you all know, π represents the ratio of a circle's circumference to its diameter. Every year on March 14, we celebrate Pi Day in recognition of the famed mathematical constant π.