# ISEE Upper Level Quantitative : Circles

## Example Questions

### 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 #1 : 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?

Explanation:

This question is asking for the length of an arc corresponding to  of a circle with radius eight feet. The question can be answered by evaluating for :

### Example Question #2 : How To Find The Length Of An Arc

Note: Figure NOT drawn to scale

Refer to the above figure.

Which is the greater quantity?

(a)

(b)

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation:

To compare  and , we note that

and

We need to be able to compare  and . If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are , respectively. However, we are not given this relationship.

### Example Question #1 : How To Find The Length Of An Arc

A giant clock has a minute hand that is six feet long. The time is now 3:50 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

The correct answer is not among these choices.

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus  of a fourth, so its tip has traveled this circumference  times.

This is

feet. This is

inches.

### Example Question #2 : How To Find The Length Of An Arc

A giant clock has a minute hand seven feet long. Which is the greater quantity?

(A) The distance traveled by the tip of the minute hand between 1:30 PM and 2:00 PM

(B) The circumference of a circle seven feet in diameter

(A) is greater

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

(A) and (B) are equal

(B) is greater

(A) and (B) are equal

Explanation:

The tip of a minute hand travels a circle whose radius is equal to the length of that minute hand, which, in this question, is seven feet long. The circumference of this circle is  times the radius, or  feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or  feet.

The circumference of a circle seven feet in diameter is  times this diameter, or  feet.

The quantities are equal.

### Example Question #5 : How To Find The Length Of An Arc

A giant clock has a minute hand four and one-half feet in length. Since noon, the tip of the minute hand has traveled  feet. Which of the following is true of the time right now?

The time is between 12:30 AM and 1:00 AM.

The time is between 11:00 PM and 11:30 PM.

The time is between 12:00 midnight and 12:30 AM.

The time is between 11:30 PM and 12:00 midnight.

The time is between 1:00 AM and 1:30 AM.

The time is between 12:00 midnight and 12:30 AM.

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle, which here is

feet.

The minute hand has traveled  feet since noon, so it has traveled the circumference of the circle

times.

Since , between 12 and  hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.