# ISEE Upper Level Quantitative : Circles

## Example Questions

### Example Question #6 : How To Find The Length Of An Arc

Acute triangle  is inscribed in a circle. Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

Explanation:

Examine the figure below, which shows  inscribed in a circle.

and

Consequently,

The two quantities are equal.

### Example Question #181 : Geometry

Which is the greater quantity?

(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.

(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

(a) A circle with radius 8 inches has crircumference  inches. An arc 10 inches long is  of that circle. , the degree measure of this arc.

(b) A circle with radius 10 inches has crircumference  inches. An arc 12 inches long is  of that circle. , the degree measure of this arc.

(a) is the greater quantity.

### Example Question #182 : Geometry

Note: figure NOT drawn to scale

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

(a)

(b)

(It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

(a) and (b) are equal

Explanation:

Since

,

the triangle is a right triangle with right angle .

is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore,  is also a semicircle, and it measures .

### Example Question #181 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Note: Figure NOT drawn to scale

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

(a)

(b) 90

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

Explanation:

The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore,

### Example Question #4 : How To Find The Angle Of A Sector

has twice the radius of . Sector 1 is part of ; Sector 2 is part of ; the two sectors are equal in area.

Which is the greater quantity?

(a) Twice the degree measure of the central angle of Sector 1

(b) The degree measure of the central angle of Sector 2

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

Explanation:

has twice the radius of , so  has four times the area of . This means that for a sector of  to have the same area as a sector of , the central angle of the latter sector must be four times that of the former sector. This makes (b) greater than (a), which is only twice that of the former sector.

### Example Question #182 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(a)

(b) 55

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Explanation:

The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, .

### Example Question #21 : Sectors

The arc-length for the shaded sector is .  What is the value of , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total circumference of the circle.

The circumference of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #7 : How To Find The Angle Of A Sector

The arc-length for the shaded sector is .  What is the value of , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total circumference of the circle.

The circumference of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #22 : Sectors

The area of the shaded sector in circle O is .  What is the angle measure , rounded to the nearest hundredth?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total area of the circle.

The area of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

### Example Question #23 : Sectors

The area of the shaded sector in circle O is .  What is the angle measure ?

˚

˚

˚

˚

˚

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total area of the circle.

The area of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.