### All ACT Math Resources

## Example Questions

### Example Question #1 : Sectors

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

**Possible Answers:**

**Correct answer:**

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 #2 : Sectors

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

What is the measure of the angle of the sector?

**Possible Answers:**

**Correct answer:**

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 #1 : Sectors

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

What is the angle measure of sector ?

**Possible Answers:**

**Correct answer:**

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 : Sectors

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

What is the angle of this sector?

**Possible Answers:**

**Correct answer:**

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 #5 : Sectors

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.

**Possible Answers:**

**Correct answer:**

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 #6 : Sectors

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

**Possible Answers:**

**Correct answer:**

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 #7 : Sectors

Figure not drawn to scale.

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

**Possible Answers:**

**Correct answer:**

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 #8 : Sectors

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

**Possible Answers:**

**Correct answer:**

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 #9 : Sectors

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

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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,