Sectors - ACT Math

You know that the area of a circle is computed by the equation:

For our data, this is:

or

Now, the sector is a percentage of the circle. For the areas, this can be represented as the fraction:

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

.

You know that the circumference of a circle is computed by the equation:

For our data, this is:

Now, the sector is a percentage of the circle. For the lengths of the circumference and the arc length, this can be represented as the fraction:

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

.

Do not overthink this question! All you need to remember is that a given circle contains degrees. This means that the sector is merely a percentage of . For our question, this percentage is , which is the same as . So, to calculate, you merely need to multiply:

This is the degree measure of the sector.

Remember that the total degree measure of a circle is . This means that if you have parts into which you have divided your circle, each spoke must be or apart.

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

To begin, you should compute the complete area of the circle:

For your data, this is:

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

Now, multiply this by the total degrees in a circle:

Rounded, this is .

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

For your data, this means,

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

Rounded to the nearest hundredth, this is .

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 .

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

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,

Circumference of a Circle =

Arc Length

A circle has a total of degrees. If our angle is located at the center and is degrees, we can do to see that our angle makes up of the complete circle.

Therefore, our arc is going to be of our total circumference.

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:

If the area of the circle is , the radius can be found using the formula for the area of a circle:

For our data, this is:

Therefore,

Now, the circumference of the circle is defined as:

For our data, this is:

Now, we know that a sector is a percentage of the total area. This percentage is easily calculated:

So, the length of the arc will merely be the same percentage, but now applied to the circumference:

The formula for the area of a sector of a circle is:

The central angle given is 120 thus: