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.

The perimeter of a given pie-piece will be equal to 2 radii plus the outer arc (which is a percentage of the circumference). For our piece, this arc will be 40/360 or 1/9 of the circumference. If the area is 361π, this means πr² = 361 and that the radius is 19. The total circumference is therefore 2 * 19 * π or 38π.

The total perimeter of the pie piece is therefore 2 * r + (1 / 9) * c = 2 * 19 + (1/9) * 38π = 38 + 38π/9

length of an arc = (degrees * 2πr)/360 = (50 * 2π * 10)/360 = 25π/9

Let's compute each quantity.

Quantity A

This is a little bit more difficult than quantity B.  It requires us to compute an arc length, for the ant does not walk around the whole pizza; however, this is not very hard. What we know is that the ant walks around , or , of the pizza.

Now, we know the circumference of a circle is  or 

For our example, let's use the latter. Since , we know:

However, our ant walks around only part of this, namely:

Quantity B

This is really just a matter of computing the circumference of the circle.  For our value, we know this to be:

Now, we can compare these by taking quantity A and reducing the fraction to be . Thus, we know that Quantity B is larger than Quantity A.

The formula for an arc length, , of a circle of a given radius, , and a given angle,  is:

Note that  is the circumference of the circle.

Conversely, for a known arc length and unknown angle, this equation can be rewritten as follows:

Plugging in the given values, it is therefore possible to find the missing angle:

To solve this problem, realize that an inscribed angle (an angle formed by two chords) is equal to twice the central angle formed by connecting the origin to the inscribed angle's endpoints:

Now, the formula for an arc length, , of a circle of a given radius, , and a given angle,  is:

Since the radius is the unknown, this equation can be rewritten as:

Plugging in our values, we find:

To solve this, notice that one piece of pizza comprises  of the total pie or (reducing)  of a pie. Now, if Susan buys five slices, she gets:

 of the pie. If the complete pie contains  calories, she will then eat the following amount of calories:

Rounding, this is  calories.

To solve this, we must ascertain the following:

1) The arc length through which the ant traveled.

2) The percentage of the total circumference in light of that arc length.

3) The percentage of 360° proportionate to that arc percentage.

To begin, let's note that the ant travelled 12 + 12 + x inches, where x is the outer arc distance. (It traveled the radius twice, remember); therefore, we know that 24 + x = 55.3 or x = 31.3.

Now, the total circumference of the circle is 2πr or 24π.  The arc is 31.3/24π percent of the total circumference; therefore, the percentage of the angle is 360 * 31.3/24π. Since the answers are approximations, use 3.14 for π. This would be 149.52°.