# ACT Math : Sectors

## Example Questions

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### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

The sector pictured above is  of the circle. What is the angle measure  for the sector?

Explanation:

A question like this is very easy. You merely need to find out what is  of the total  degrees in a circle. This is:

. That is it!

### Example Question #1 : Sectors

The area of sector  is . This figure is not drawn to scale.

What is the measure of the angle of the sector?

Explanation:

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:

.

### Example Question #3 : How To Find The Angle For A Percentage Of A Circle

The arc length of sector above is . This figure is not drawn to scale.

What is the angle measure of sector ?

Explanation:

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:

.

### Example Question #4 : How To Find The Angle For A Percentage Of A Circle

Sector  is  of the total circle. This figure is not drawn to scale.

What is the angle of this sector?

Explanation:

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.

### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

A bike wheel has  evenly spaced spokes spreading from its center to its tire. What must the angle be for the spokes in order to guarantee this even spacing? Round to the nearest hundredth.

Explanation:

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.

### Example Question #1 : Sectors

The sector above represents what percentage of the complete circle? Round to the nearest hundredth.

Explanation:

This question is very easy to solve. You merely need to find what percentage  degrees is in comparison with the whole of  degrees for the circle. This is:

or  or

Rounding, you get .

### Example Question #1 : Sectors

Figure not drawn to scale.

What percentage of the circle is sector ? Round to the nearest hundredth of a percent.

Explanation:

Do not overthink this question! All you need to remember is that a given circle contains  degrees. This means that your sector percentage is merely a ratio of the angle  to . This is:
or

Rounded, this is .

### Example Question #3 : Sectors

A sector of a circle is bounded by the minor arc of . What percentage of the circle is the arc?

Explanation:

The only information provided is that the sector is . When finding a percent, it's helpful to remember that percents are a clean way to show fractions—that is, portions—of a whole.

Therefore, in order to solve for what percent the sector represents of the entire circle, we need to find out what fraction the sector makes up of the entire circle. Keep in mind that a circle measures up to . This means that the sector is  of the total . This can be written out as:

This can be simplified to:

Note that the degrees signs are gone. This is because when you divide degrees by degrees, the units cancel out! This simplified fraction means that  was one-fifth of the entire circle. Now, this fraction may be turned into a percent.

In order to go from fractions to percents, it's helpful to keep in mind that decimals are the important intermediate step. That is, as soon as a fraction can be turned into decimal form, it can easily be converted into a percent by multiplying by 100.

That is, the sector makes up  of the entire circle.

### Example Question #4 : Sectors

A sector of an angle is bounded the major arc of . What percent of the circle does this make up?

Explanation:

The only information provided is that the sector is . When finding a percent, it's helpful to remember that percents are a clean way to show fractions—that is portions—of a whole. In order to solve for what percent the sector makes of the entire circle, we need to find out what fraction the sector makes up of the entire circle.

Keep in mind that a circle measures  degrees. This means that the sector is  of the total . This can be written out as:

This can be simplified to:

Note that the degrees signs are gone. This is because when you divide degrees by degrees, the units cancel out! This simplified fraction means that  was three-fourths of the entire circle. Now, this fraction may be turned into a percent.

In order to go from fractions to percents, it's helpful to keep in mind that decimals are the important intermediate step. That is, as soon as a fraction can be turned into decimal form, it can easily be converted into a percent by multiplying by .

That is, the sector makes up  of the entire circle.

### Example Question #1 : Sectors

In circle , central angle  and inscribed angle  both intercept minor arc . If , what is the measure of