Question

Find the circumferences of the two circles circle a has a radius of 21 meters and circle b has a radius of 28 meters Is the relationship between the radius of a circle and the distance around the circle the same for all circles

Answer 1

The circumference for circle a is
`\\ C = 131.9430`m.

The circumference for circle b is
`\\ C = 175.9240`m.

The relationship between the radius of a circle and the circumference (the distance around the circle) is constant and is the same for all circles and can be written as
`\\ (C)/(r) = 2\pi` or, in a less familiar form,
`\\ (r)/(C) = (1)/(2\pi)`. The number
`\\ \pi` is constant for all circles and has infinite digits,
`\\ \pi = 3.14159265358979....`.

Explanation:

The circumference of a circle is given by:

`\\ C = 2*\pi*r` [1]

Where

`\\ C` is the circle's circumference.

`\\ r` is the radius of the circle.

And

`\\ \pi = 3.141592....` is a constant value (explained below)

We can say that the distance around the circle is the circle's circumference.

The circumferences of the two circles given are:

Circle a, with radius equals to 21 meters (
`\\ r = 21m`).

Using [1], using four decimals for
`\\ \pi`, we have:

`\\ C = 2*\pi*r`

`\\ C = 2*3.1415*21`m

`\\ C = 131.9430`m

Then, the circumference for circle a is
`\\ C = 131.9430`m.

Circle b, with radius equals to 28 meters (
`\\ r = 28m`).

`\\ C = 2*3.1415*28`m

`\\ C = 175.9240`m

And, the circumference for circle b is
`\\ C = 175.9240`m.

We know that

`\\ 2r = D`

That is, the diameter of the circle is twice its radius.

Then, if we take the distance around the circle and we divided it by
`\\ 2r`

`\\ (C)/(2r) = (C)/(D) = \pi`

This ratio, that is, the relationship between the distance around the circle (circumference) and the diameter of a circle is
`\\ \pi` and is constant for all circles. This result is called the
`\\ \pi` number, which is, approximately,
`\\ \pi = 3.141592653589793238....` (it has infinite number of digits).

We can observe that the relationship between the radius of a circle and the circumference is also constant:

`\\ (C)/(2r) = (C)/(D) = \pi`

`\\ (C)/(2r) = \pi`

`\\ (C)/(r) = 2\pi`

However, this relationship is
`\\ 2\pi`.

We can rewrite it as

`\\ (r)/(C) = (1)/(2\pi)`

And it is also constant.

Answer 2

Circle A is 42π

circle C is 56π