# ISEE Upper Level Quantitative : Sectors

## Example Questions

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

What is the angle measure of  in the figure above if the sector comprises 37% of the circle?

˚

˚

˚

˚

˚

˚

Explanation:

It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply  by ˚.  This yields ˚.

### Example Question #2 : How To Find The Angle For A Percentage Of A Circle

What is the angle measure of  in the figure above if the sector comprises % of the circle?

˚

˚

˚

˚

˚

˚

Explanation:

It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply  by ˚.  This yields ˚.

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

What is the angle measure of  in the figure if the sector comprises  of the circle?

˚

˚

˚

˚

˚

˚

Explanation:

It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply  by ˚.  This yields ˚

### 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

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

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

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 #2 : 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?

Explanation:

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 #3 : 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?

Explanation:

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 : How To Find The Area Of A Sector

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

(a) The area of

(b) The area of the orange semicircle

(a) is the greater quantity

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

Explanation:

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 : How To Find The Area Of A Sector

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

(a) The area of the orange semicircle

(b) The area of

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Explanation:

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 #6 : 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.

Explanation:

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

.

### Example Question #2 : Sectors

The clock at the town square has a minute hand eight feet long. How far has its tip traveled since noon if it is now 12:58 PM?