ISEE Upper Level Quantitative : Circles

Example Questions

Example Question #5 : How To Find The Area Of A Circle

The radius of a circle is . Give the circumference of the circle in terms of .      Explanation:

The circumference of a circle is times its radius. Therefore, since the radius is , the circumference is Example Question #6 : How To Find The Area Of A Circle

The areas of six circles form an arithmetic sequence. The second-smallest circle has a radius twice that of the smallest circle.

Which is the greater quantity?

(a) The area of the largest circle.

(b) Twice the area of the third-largest circle.

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

(b) is greater

(a) is greater

(a) and (b) are equal

(b) is greater

Explanation:

Let be the radius of the smallest circle. Then the second-smallest circle has radius . Their areas, respectively, are and The areas form an arithmetic sequence, so their common difference is .

The six areas are The third-largest circle has area ; twice this is . This is greater than the area of the largest circle, which is . (b) is the greater quantity.

Example Question #21 : Circles

The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?

(a) The area of the largest circle

(b) Twice the area of the third-smallest circle

(b) is greater

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

(a) and (b) are equal

(a) is greater

(a) is greater

Explanation:

Call the radius of the smallest circle . The radius of the second-smallest circle is then , and the common difference of the radii is .

The radii of the six circles are, from least to greatest: The largest circle has area The third-smallest circle has area: Twice this is The area of the sixth circle is greater than twice that of the third-smallest circle, so the correct choice is that (a) is greater.

Example Question #1 : How To Find The Area Of A Circle In the above figure, .

Which is the greater quantity?

(a) Twice the area of inner gray ring

(b) The area of the white ring

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula :    The white ring has as its area the difference of the areas of the second-largest and third-largest circles: The inner gray ring has as its area the difference of the areas of the third-largest and smallest circles: .

Twice this is , which is greater than the area of the white ring.

Example Question #9 : How To Find The Area Of A Circle In the above figure, .

Which is the greater quantity?

(a) Six times the area of the white circle

(b) The area of the outer ring

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

(b) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula :    The outer gray ring is the region between the largest and second-largest circles, and has area Six times the area of the white (inner) circle is , which is less than the area of the outer ring, .

Example Question #1 : How To Find Circumference

A giant clock has a minute hand five yards in length. Since noon, the tip of the minute hand has traveled feet. Which is the greater quantity?

(A) The amount of time that has passed since noon

(B) The amount of time until midnight

(A) and (B) are equal

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

(B) is greater

(A) is greater

(A) is greater

Explanation:

Five yards is equal to fifteen feet, which is the length of the minute hand. Subsequently, fifteen feet is the radius of the circle traveled by its tip in one hour; the circumference of this circle is times this, or feet.

In one six-hour period, the minute hand revolves six times, so its tip travles six times the circumference, or The clock has traveled farther than this, so the time is later than 6:00 PM, and more time has elapsed since noon than is left until midnight. This makes (A) greater.

Example Question #2 : How To Find Circumference

Compare the two quantities:

Quantity A: The area of a circle with radius Quantity B: The circumference of a circle with radius The quantity in Column A is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

Explanation:

Recall for this question that the formulae for the area and circumference of a circle are, respectively:  For our two quantities, we have:

Quantity A: Quantity B: Therefore, quantity A is greater.

Example Question #3 : How To Find Circumference

The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.

3 feet 10 inches

1 foot, 11 inches

2 feet

2 feet, 1 inch

4 feet 2 inches

1 foot, 11 inches

Explanation:

The circumference of a circle can be determined by multiplying its radius by , so the circumferences of the two smallest circles are and The circumferences form an arithmetic sequence with common difference The circumference of a circle can therefore be found using the formula where and ; we are looking for that of the th smallest circle, so    Since the radius of a circle is the circumference of the circle divided by , the radius of this eighth circle is inches, or 1 foot 11 inches.

Example Question #4 : How To Find Circumference The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.

Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?      Explanation:

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is feet.

The distance around the track is about feet.

Diane wants to run two miles, or feet. circuits around the track.

Equivalently, she will run the track 5 complete times for a total of about feet,

so she will have feet to go.

She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving feet.

She will almost, but not quite, finish the 628 feet from Point D to Point B.

The correct response is Point B.

Example Question #161 : Geometry The track at Monroe Elementary School is a perfect circle of radius 400 feet, and is shown in the above figure.

Evan and his younger brother Mike both start running from Point A. Evan runs counterclockwise, running once around the track and then on to Point E; Mike runs clockwise, meeting Evan at Point E and stopping.

Which of the following is the greater quantity?

(a) Twice Mike's average speed.

(b) Evan's average speed.

(Assume the five points are evenly spaced)

(b) is greater

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

(a) and (b) are equal

(a) is greater

It is not actually necessary to know the radius or length of the track if we know the points are equally spaced. Evan runs once around the track counterclockwise and then on to Point E, which is the next point after A; this means he runs around the track times. Mike runs around the track clockwise from Point A to Point E, in the same time, meaning he runs around the track times.
Therefore, Evan's speed is times Mike's speed. As a result, Twice Mike's speed would be greater than Evan's speed, making (b) the greater. 