### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Sectors

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a sector of Circle A

(b) The area of Circle B

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) and (b) are equal.

Let be the radius of Circle B. The radius of Circle A is therefore .

A sector of a circle comprises of the circle. The sector of circle A has area , the area of Circle B. The two quantities are equal.

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

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

**Possible Answers:**

**Correct answer:**

To find the area of a sector, you need to find a *percentage* of the *total* area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

Now, we will multiply this by the *total* area of the circle. Recall that we find such an area according to the equation:

For our problem,

Therefore, our equation is:

Using your calculator, you can determine that this is approximately .

### Example Question #1 : Sectors

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

**Possible Answers:**

**Correct answer:**

To find the area of a sector, you need to find a *percentage* of the *total* area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

Now, we will multiply this by the *total* area of the circle. Recall that we find such an area according to the equation:

For our problem,

Therefore, our equation is:

Using your calculator, you can determine that this is approximately .

### Example Question #4 : Sectors

Refer to the above figure, Which is the greater quantity?

(a) The area of

(b) The area of the orange semicircle

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

(a) is the greater quantity

has two angles of degree measure 45; the third angle must measure 90 degrees, making a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so . The area of a right triangle is half the product of its legs, so

Also, if , then the orange semicircle has diameter 1 and radius . Its area can be found by substituting in the formula:

has a greater area than the orange semicircle.

### Example Question #5 : Sectors

Refer to the above figure, Which is the greater quantity?

(a) The area of the orange semicircle

(b) The area of

**Possible Answers:**

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

has two angles of degree measure 60; its third angle must also have measure 60, making an equilateral triangle

For the sake of simplicity, let ; the reasoning is independent of the actual length. The area of equilateral can be found by substituting in the formula

Also, if , then the orange semicircle has diameter 1 and radius . Its area can be found by substituting in the formula:

has a greater area than the orange semicircle.

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

The above circle, which is divided into sectors of equal size, has diameter 20. Give the area of the shaded region.

**Possible Answers:**

**Correct answer:**

The radius of a circle is half its diameter; the radius of the circle in the diagram is half of 20, or 10.

To find the area of the circle, set in the area formula:

The circle is divided into sixteen sectors of equal size, five of which are shaded; the shaded portion is

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