Should have added the 0 to make it more clear the decimal point.

0.667 if you round correctly

Or 0.666666666...

Isnt C = 2piR the circumference equation?

It is

Pi is not used.

It is just a triangle.

Why would you use a triangular calculation to calculate a curve?

Because it is easy way to confirm the 0.666666... feet per miles squared. I only did it a few times. It is something like 0.6661 or something like that. I just said 0.666 because that is a symbol and it is very close to it. I would never care about the thousandth I could be off in the calculation.

You know that that formula describes a parabola and not a circle?

lmao you used the Pythagorean theorem

Yes but I want a triangle not a circle.

I don't even know how that relates to the curvature of the earth

That parabolic equation is known to be false and using the proper (Sine) formula you can get it closer

If you assume the Earth is perfectly flat all that formula would give you is the distance between the point you're looking at and whatever distance you set as the radius

Or use a readymade calculator like this https://www.metabunk.org/curve/

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

So whatever point you were looking at compared to an unspecified point below you

don't know how this is mildly related to the curvature of the earth

You go down to the center of the earth. The radius it is called. You go out for however many miles it is. Use google earth or something to see. So two sides of a triangle at a right angle. Then C is the distance from the center of the earth up to meet the line of sight. That will give you the curvature you should see.

line of sight is not perpendicular

lol no you don't use the Pythagorean theorem to find the curvature of the earth

It would absolutely be. The curve would be below the line of sight.

h = r * (1 - cos a) is accurate for any distance

@mineyful Yes doing that would prove the earth is a ball. Don't do that. Is that what you are saying?

The Earth's radius (r) is 6371 km or 3959 miles, based on numbers from Wikipedia,
which gives a circumference (c)of c = 2 * π * r = 40 030 km

We wish to find the height (h) which is the drop in curvature over the distance (d)

Using the circumference we find that 1 kilometer has the angle
360° / 40 030 km = 0.009°. The angle (a) is then a = 0.009° * distance (d)

The derived formula h = r * (1 - cos a) is accurate for any distance (d)

Why should we not show that the earth is a sphere?

@mineyful Do you not understand almost no one would understand how calculus is used to get that formula. Few take college calculus. However most people in High School take math and would understand Pathagriums theory.

My son knows very little about math. But he knows that.

that's trig

Also I'm re-reading some of the calculations you showed earlier. If we took your .666 feet per mile figure you would see just 6.66 feet of curve over 10 miles not 66.6

True but the formula requires using calculus to come up with that formula.

If someone does not trust anything they just read about. Like that formula. Well then they dismiss it. Will not believe it is valid.

it shows you the derivation of the formula if you just read what I said

But they will understand what that simple formula that can be used in many useful ways.

the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

what I posted is not calculus

I am not questioning the validity of that formula at all. I just took Calculus so understand how it was invented.

"True but the formula requires using calculus to come up with that formula."

that's not correct

OK explain to me how they came up with it.