Since yesterday was Tau Day, it got me thinking about a similar issue with the measurements of a square.
If you read the post referenced above, you'll have seen something called The Tau Manifesto, which says that pi is the wrong number to use in equations dealing with circles, and that the correct number to use is tau, which is equal to 2 * pi.
And now I'm saying the same thing for a square...that we shouldn't be using the side as the unit of measurement but we should be using the apothem instead.
I've had this idea for awhile, but I didn't know the word that I needed to use until yesterday, so I'd been using the letter h for the idea of a half side.
So, what is an apothem? According to Wikipedia:
The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides.
As you can see, the apothem is equal to 1/2 the length of a side of the square,
The (incorrect) measurements for the perimeter and ares of a square, as well as the volume of a cube are:
Perimeter of a square:
P = 4s (where s is the length of a side of the square)
Area of a square:
A = s^2 (s squared)
Volume of a cube:
V = s^3 (s cubed)
These equations do provide technically correct answers but they don't work in calculus.
Let's look at the equations of a circle (we'll use pi for now, since most of us are familiar with those equations).
If we imagine a circle as being a regular polygon made up of an infinite number of sides and that the radius, r, is the apothem we end up with the following equations:
Circumference (perimeter) of a circle:
C = 2 * pi * r (2 pi r)
Area of a circle:
A = pi * r^2 (pi r squared)
Volume of a sphere:
V = 4/3 pi * r^3 (four thirds pi r cubed)
Ignoring the volume for now, we'll take the derivative of the area, which should give us the circumference:
If A = pi * r^2, then the derivative is dA/dr = 2 pi * r...using the power rule we reduce the exponent from 2 to 1 and multiply the result by 2, and indeed the answer is the circumference.
But if we do the same thing with a square:
A = s^2, using the power rule we get dA/ds = 2s, which is not equal to the perimeter of 4s!
If we use the apothem instead, we have:
s = 2a, then
P = 8a and A = 4a^2 and now dA/da = 2 * 4a or 8a, which is the perimeter!
If you're using the apothem you can find the area of a square by multiplying the semi-perimeter (sp) by the apothem as follows:
A = sp * a or A = 1/2 * p * a, which is the same equation used in a circle:
A = 1/2 * c * r, where c = 2 * pi * r, so 1/2 * c = pi * r, therefore A = pi * r * r or pi * r^2.
Now, to get to the volume of a sphere, we need to take an intermediate step and find the surface area of the sphere first which is four times the area of a circle (think of the cover of a baseball):
SA = 4 * pi * r^2 (4 pi r squared)
Now, to find the volume we'll use the power rule in reverse: raise the exponent from 2 to 3 and divide the result by 3:
V = 4/3 * pi * r^3
To find the surface area of a cube, you multiply the area of each square (4a^2) by 6 (the number of sides of a cube):
SA = 6 * (4a^2) which equals 24a^2. To find the volume of the cube, we use the power rule in reverse and we have:
A = 8a^3 (because we raised the exponent in the SA equation from 2 to 3 and divided the result by 3).
If we go back to our original (incorrect) volume formula, we have:
V = s^3, but if we replace s with 2a we now have:
V = (2a)^3 or 8a^3!
So, now you know why the apothem is a better measure for the equations of a square that the side is.
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