If an ecologist is measuring species composition using a circular region with an area of , what is the diameter of the region?

Recall the formula for the area of a circle:

We also know that r, the radius, is half the length of the diameter. Therefore, if we can find the radius, we can find the diameter:

In order to calculate the ratio of circumference to diameter, we need an equation that involves both variables. The formula for circumference is as follows:

We need to express the radius in terms of diameter. The radius of a circle is half of its diameter, so we can rewrite the formula as:

If we divide both sides by the diameter, on the left side we will have , which is the ratio of circumference to diameter:

So, for any circle, the ratio of its circumference to its diameter is equal to , which is actually the definition of this very important mathematical constant.

To find the ratio between the diameter and the circumference of a circle, we need to use the formula for the circumference of a circle:

We can see this formula is in terms of radius, so we need to express it in a way that the circumference is in terms of diameter. Using the knowledge that the radius is half of the diameter:

Now that we have a simple formula involving the circumference and diameter, we can see that we will have the ratio of diameter to circumference if we divide both sides by the circumference. We then divide both sides by  to isolate the ratio of diameter to circumference and find our solution:

To calculate the ratio of the diameter to the circumference of the square we should first get the diameter, which is the same as the length of a side of the sqaure. To do so we just need to take the square root of the area of the square, which is 4. Also we should remember that the circumference is given by , where  is the length of the radius.

Now we should notice that this formula can also be written .

The ratio we are looking for is . Therefore, this ratio will always be  and this is our final answer.

The perimeter of the circle allows us to find the side of equilateral triangle ABC;  or 2.

From there, we can also find the length of the height of equilateral triangle ABC with the formula , which turns out to be .

Since in an equilateral triangle, the center of gravity is at  from any vertex, it follows that the radius of the circle is . 

For a given circle of circumference  and radius , .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as

 . 

Therefore, the ratio of the circumference to the diameter of this and all other circles is .

For a given circle of circumference  and radius , .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as 

. 

Therefore, the ratio of the circumference to the diameter of this and all other circles is,

 .

For a given circle of circumference  and radius , .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as, 

. 

Therefore, the ratio of the circumference to the diameter of this circles is,

 .

The circumference of a circle is given by 

where  

We are told the diameter so we just need to plug in our value into the equation: