This question is easy on the whole, though you must not be intimidated by one small fact that we will soon see. Set up your standard circumference equation:

The circumference is  feet, so we can say:

Solving for , we get:

Some students may be intimidated by having  in the denominator; however, there is no need for such intimidation. This is simply the answer!

Since we know that the area of Square  is , we know , where  is the length of one of its sides. From this, we can solve for  by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that  is . By careful guessing, you can quickly see that  is . From this, you know that the diameter of your circle must be half of , or  (because it is circumscribed). Therefore, you can draw:

The diameter is

To solve for the largest diameter multiply each side by 2.  

The resulting formula for diamenter is

 .  

Substitute in 5  for S and solve. Diameter =  = 2.89 cm 

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

Therefore, if , . Therefore, Quantity B is larger.

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, .  Since , we know:

Quantity B

This is very easy.  Recall that:

Therefore, if , .  

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that  is . Therefore, .  This means that . Therefore, Quantity B is larger.

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

or

The largest sector thus has an area equal to

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for ,  or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

We can set up a ratio to calculate the angle measure as such: 6/25 = x/360, since there are 360 degrees in a circle.  Solving, we obtain x = 86.4 degrees. 

First of all, calculate the angle of each of the full pieces of pie. This is easily found:

Though small, this is what the information tells us! Now, we know that if two people are eating the piece, with one having twice the amount of the other, the angles must be  for the smaller piece and  for the larger one. Thus, we can write the equation:

Simplifying, we get:

That is a tiny piece, but that is what is called for by the data!

For each of these, you need to compute the total measurement applicable to the given data. For Quantity A, this will be the total circumference. For Quantity B, this will be the total area. You will then divide the given sector calculation by this total amount. By multiplying this percentage by , you will find the degree measure of each; however, you will merely need to stop at the percentage (since both are percentages of the same number, namely ).

Quantity A

The total circumference is calculated using the standard equation:

 or, for our data: 

Thus, our pertinent percentage is:

Quantity B

For this, the area is computed by the formula:

 or, for our data: 

Thus, our percentage is:

Clearly, , so A is larger than B.