Why do we say pi is the ratio of the circumference to the diameter, and not diameter to the circumference?

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Everywhere I see it written, $\pi$ is described as "the ratio of the circumference to the diameter". I know $\pi = C/d$, but the ratio $3.14...:1$ is $3.14$... diameters to $1$ circumference. So shouldn't it be described as the ratio of diameter to circumference?

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You're almost there in your reasoning. However, you're treating the terms diameter and circumference as UNITS in your explanation. This is odd because the diameter is not a unit; nor is the circumference. They are scalar measurements of some other unit, such as centimeters, for example.

If you have a diameter of 100cm, then the circumference would be about 314 centimeters. We do not apply the terms diameter and circumference in our ratio. To do so is confusing.

But let's look at your reasoning a little further. How many diameters make up one circumference? When you look at this question and answer it straight out, you would say approximately 3.14 diameters make up 1 circumference as you described in your question. We have two dissimilar units in this ratio, which makes it meaningless.

(3.14 diameters)/(1 circumference)

... this is actually equal to 1, because both lengths are equal. Your problem is that you're comparing ideas rather than scalar values.

Simply stated, the length of the circumference is about 3.14 times a long as the diameter. Therefore the ratio of these two scalar lengths is 3.14... we know this as pi. And these measures are given specific linear values with a common unit that is linear, not descriptive.