Hey folks, throughout all of my schooling, I was taught to use pi (π) to solve equations involving circles and the like, but never once has a teacher even tried to explain why. I was taught to remember formulas...
Circumference = 2 * pi * radius
Area = pi * radius²
We'll use a circle with a diameter of 10 units, and a square with a leg length of 10 units as our example(keep things simple).
Example(π formulas):
Area = pi * radius²
Area = 3.14 * 5²
Area = 3.14 * 25
Area = 78.5 units²
Circumference = 2 * pi * radius
Circumference = 2 * 3.14 * (diameter/2)
Circumference = 2 * 3.14 * 5
Circumference = 6.28 * 5
Circumference = 31.4 units
My distaste for formulas and memorization led me to a little research, and because I enjoy messing with my under-qualified teachers, I came up with my own formulas(bonus: I find them to be simpler). Here I will try to convey my understanding.
Pi ≈ 3.14
It was derived through calculations beyond me but this picture helped me to understand.
Pi = 4 * the area a ¼ circle fills of a ¼ square having the same width/diameter
(4 * The shaded grey square minus what's outside the circle)
I took this info and did some math. I took pi and divided by 4, because the fourth section(see pic) was multiplied by 4 to become pi. In other words Pi is 4 times the percentage a quarter circle occupies of an encompassing square. This is where I believe radians we're derived. As a full 360° circle is equal to 2π radians (note:a full circles circumference is 2π * radius)
Dividing pi by 4 gave me ≈ 0.785. This number is the percentage a full circle fills of a square with the same dimensions, and because of the nature of this propurtionality the same percentage can be applied to circumference, as in 0.785 is also the percentage a circles circumference is of it's encompassing squares' perimeter.
Let's make it a percentage, %78.5 and call it π%.
Area = π*r²
Radius² gives you the area of a squares quadrant. Pi then multiplies it by 4(4 quadrants) minus the area the circle does not fill of a quadrant (3.14159...)
Using π%(%78.5) you simply multiply it by the area of the square(diameter²).
Example(π% formulas):
Area = π% * d²(or length * width for ovals)
Area = %78.5 * 10²
Area = 0.785 * 100
Area = 78.5 units²
Circumference = π% * (4 * diameter)
Circumference = %78.5 * (4 * 10)
Circumference = 0.785 * 40
Circumference = 31.4 units
I know this is confusing, but it has simplified equations of this nature for me, and hopefully you! If you have any questions, corrections, or would like me to expound upon anything please leave a comment and I'll gladly respond appropriately. Thanks for reading!