Why is the ratio of a circle’s circumference to its diameter always pi?

Question

Why is the ratio of a circle’s circumference to its diameter always pi?

Are there exceptions to this?

Answer 1

As has been said in other answers, the number π is often defined as π=C/d, where C is the circumference of a circle and d is the diameter of a circle. Your questions seems to imply: "A circle? But how do we know that this is the same for all circles?" That's a good question! The reason is that all circles are similar. Just as the ratios of sides in similar triangles are the same, the ratio of a circle's circumference to its diameter is always π

I am not "proving" this fact, because a proof depends on what assumptions you start with. Since you asked if there are any exceptions, it is worth mentioning an interesting result. In the usual system of Euclidean geometry, π is always one constant number, as discussed above. But in other systems of geometry, it may have other values. If you are interested, there is a wonderful article on this topic by Cornelia Van Cott: https://www.maa.org/sites/default/files/pdf/awards/mathhorizons.23.3.24.pdf

Answer 2

Well, the circumference scales exactly with the diameter. So, no matter how large or small the circle is, the ratio will be the same.

Answer 3

Look at the formula for the circumference of a circle

C = pi • d

C = circumference

d = diameter

If you multiply the diameter by any number

the circumference will also be multiplied by the same number. For any circle if you divide the circumference by the diameter you always get pi

Answer 4

circumference = π x diameter for all circles.

Answer 5

Please start with the definition of ∏:

∏ = (Circumference of a circle) / (diameter of the circle)

Since this is the definition of ∏, there are no exceptions.

A symbol for the ratio of a circle's circumference to its diameter was needed because this ratio appears very often in math . The Greek letter ∏ (pi) was chosen.

NOTE: Using C=∏d or C=2∏r to show that this is always true is simply applying the definition.