# ACT Math : How to find the area of a sector

## Example Questions

### Example Question #1 : Sectors

The radius of the circle above is  and .  What is the area of the shaded section of the circle?

Explanation:

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

### Example Question #511 : Act Math

Stan is making some giant circular pies for his friend's wedding. The pies each have a diameter of  and will be cut into  equal pieces. Stan wants to have three pieces to himself. What is the surface area of the pie that Stan will eat if he eats three pieces?

Explanation:

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.

### Example Question #512 : Act Math

If a circle, , is centered at the origin, and has an area of , what is the area of the sector defined by the lines  and the -axis?

Explanation:

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. .

### Example Question #511 : Act Math

A pizza recipe requires  drops of hot sauce for every   of surface area. If these drops are evenly distributed, what is the amount of hot sauce on a piece of pizza that is   in diameter, cut into  pieces? Round to the nearest drop. Assume that the sauce is evenly spread.

Explanation:

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:

### Example Question #1 : How To Find The Area Of A Sector

On a circle with a radius of , what is the area of a sector having an arc length of ? Round to the nearest hundredth.

Explanation:

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:

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 .

### Example Question #1 : How To Find The Area Of A Sector

Find the area of a 90 degree sector of a circle whose radius is 2.

Explanation:

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,

### Example Question #1 : How To Find The Area Of A Sector

A group of students wish to cut a circular pizza of diameter  into equal slices, each with an angle of . What is the total surface area, in square inches, of each slice of pizza? Round to the nearest hundredth.

25.81

19.41

34.14

22.34

89.36

22.34

Explanation:

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: