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

Answer:

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 $\\ \frac{C}{r} = 2\pi$ or, in a less familiar form, $\\ \frac{r}{C} = \frac{1}{2\pi}$. The number $\\ \pi$ is constant for all circles and has infinite digits, $\\ \pi = 3.14159265358979....$.

Step-by-step 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 $\\ 2r$

$\\ \frac{C}{2r} = \frac{C}{D} = \pi$

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:

$\\ \frac{C}{2r} = \frac{C}{D} = \pi$

$\\ \frac{C}{2r} = \pi$

$\\ \frac{C}{r} = 2\pi$

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

We can rewrite it as  

$\\ \frac{r}{C} = \frac{1}{2\pi}$

And it is also constant.