The area  of a circle is defined by , where  is the radius of the circle. We are provided with the diameter  of the circle, which is twice the length of .

If , then 

Therefore:

Our first step will be to draw two radii, from the circle's center to two adjacent vertexes of the square, forming a triangle, the third side of which is the edge of the square. Because these radii bisect the square's right angles, we can determine that the triangle we drew is a 45˚ - 45˚ - 90˚ right triangle. Because of the rules governing those isoceles right triangles, we can then determine that, because the two legs of the triangle are 5 (radius = 5), the hypotenuse (the square's side) must equal .

Now we can find the square's area:

and the circle's area:

Finally we subtract the square's area from the circle's area:

An engineer is designing a circular hatch for a submarine. If the hatch must have a circumference of , what will its area be?

To find area, we will need the radius. We can find the radius using the following formula:

So, plug in and solve for r

Next, use the area formula to find the area:

So we have our answer!

The area and the diameter of a circle are related by the formula

Substituting:

If , then

,

and  varies inversely as the fourth power of .

The area of a circle is found using the following formula:

Set :

If the pitcher plant Sarracenia purpurea has a circular opening with a circumference of 6 inches, what is the area of the opening?

We need to work backward from circumference to find area.

Circumference can be found as follows:

Use this to find "r" which we will use to find the area:

Next, find area using the following:

Let  be the sidelength of the square is the square and  be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

We find the ratio of  to  - that is,  - as follows:

The correct ratio is .

Let  be the radius of circle C. Then Circle B has radius equal to twice this, or , and Circle A has radius equal to three times that, or . The circumference of a circle is equal to  times the radius, so the circumferences of the three circles are:

A: 

B: 

C: 

These circumferences add up to , so set up and solve the equation:

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

Plug in our points and simplify:

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

A close reading of the question reveals that we are given the circumference and asked to find the radius.

Circumference formula:

So, 

So 17 feet