A central angle and the arc it intercepts always have the same angle, and an inscribed angle always has an angle half the measure of the arc it intercepts. Thus, if the central angle measures , the corresponding inscribed angle must measure .

Therefore, 

Area of Circle = πr² = π4² = 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

To begin with, we need to break down the word problem to figure out exactly what is being asked of us. We are given a diameter and some clues about how much pie Stan wants to eat. 

To solve this problem, we will go through a few steps. First, we have to find the area of a whole pie. Then, we have to find the area of one slice of pie. Finally, we have to multiply the area of a slice of pie by , since Stan wants to eat three slices of pie, not just one.

To find the area of a whole pie, we will need to recall the formula for area of a circle:

In this case, we aren't given , but we do know , which is equal to . Since , . Substituting  for into the equation for an area of a circle, we get:

Each of the answer choices includes , so don't change  into a decimal.

Next, to find the area of one slice of pie, we want to multiply the total area of the circle by the fractional area we are interested in.

We are told in the question that one slice is equivalent to  of the pie since each pie will be cut into  slices. We also know that Stan wants to eat three slices of pie. Therefore, we will need to multiply the total area by .

So, to find the total area of pie that Stan wants to eat, we will perform the following calculation:

So, Stan wants to eat  of pie.

Our first step is to find the angle between the lines  and the x-axis. Because one of our lines is the x-axis, we can do this by simply taking the inverse-tangent of the slope of . We know the slope of  is equal to 1. 

 radians or  degrees. There are  radians or  degrees in a circle. Dividing our angle by the whole, we see that our angle is  of a complete circle.

Therefore, the area of our sector must be  of the total area of the circle. .

To start, calculate the total area of the pizza. This is easily done, using the equation for the area of a circle:

Be careful, though. Notice that they give you the diameter, so the radius is :

Now, if this is cut into eight pieces, this would be:

 or approximately   per piece.

Now, you need to divide this by  to find out the drop amount:

For your answer, you will round to .

To solve this, you need to find the percentage of the given arc with respect to the total circumference. The total circumference of your circle is found merely by using:

 or, for your data 

So, your arc is the following percentage of the circle:

 or approximately 

Now, the total area of your circle is:

 or 

Your sector will be merely  times  or .

To solve, first find area. Substitute 2 in for the radius.

Then, to find the area of a sector, divide 90 by the total degrees in a circle and multiply it by the area.

Thus,

A group of students wish to cut a circular pizza of diameter   into pieces with an angle of  degrees. What is the total surface area of pizza that each student will recieve if every student eats two pieces of pizza?

To solve a question like this, you need to compute the percentage of the circle that will be represented by a cut of  degrees. The total equation for this is:

The total area is found merely by using the standard equation for the area of a circle:

For your data, this is:

Multiply this by  and you get approximately  

Circumference of a Circle = 

Arc Length

Recall that the length of an arc is merely a percentage of the circumference. The circumference is found by the equation:

For our data, this is:

Now the percentage for our arc is based on the angle  and the total degrees in a circle, namely, .

So, the length of the arc is: