Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Circles

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

The radius of a circle is x - 5 . Give the circumference of the circle in terms of .

Possible Answers:

$\pi x^{2} -10 \pi x+ 25 \pi$

$4 \pi x^{2} -20 \pi x+ 50 \pi$

$2 \pi x^{2} -20 \pi x+ 50 \pi$

$2 \pi x - 10\pi$

$\pi x - 5\pi$

Correct answer:

$2 \pi x - 10\pi$

Explanation:

The circumference of a circle is $2 \pi$ times its radius. Therefore, since the radius is x - 5 , the circumference is

$2 \pi \left (x - 5 \right ) = 2 \pi x - 2 \pi \cdot 5 = 2 \pi x - 10 \pi$

Report an Error

Example Question #1 : 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.

Possible Answers:

(b) is greater

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

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

Let be the radius of the smallest circle. Then the second-smallest circle has radius . Their areas, respectively, are

$A_{1} = \pi r ^{2}$

and

$A_{2} = \pi \left (2r \right )^{2} = 4 \pi r^{2}$

The areas form an arithmetic sequence, so their common difference is

$4 \pi r^{2} - \pi r^{2} = 3 \pi r^{2}$ .

The six areas are

$\pi r ^{2}, 4 \pi r ^{2}, 7 \pi r ^{2}, 10 \pi r ^{2}, 13 \pi r ^{2}, 17 \pi r ^{2}$

The third-largest circle has area $10 \pi r ^{2}$ ; twice this is $20 \pi r ^{2}$ . This is greater than the area of the largest circle, which is $17\pi r ^{2}$ . (b) is the greater quantity.

Report an Error

Example Question #6 : How To Find The Area Of A Circle

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

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

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

Correct answer:

(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 2r - r = r .

The radii of the six circles are, from least to greatest:

r, 2r, 3r, 4r, 5r, 6r

The largest circle has area

$A_{6} = \pi (6r)^{2} = \pi \cdot 36 r^{2} = 36 \pi r^{2}$

The third-smallest circle has area:

$A_{3} = \pi (3r)^{2} = \pi \cdot 9 r^{2} = 9 \pi r^{2}$

Twice this is

$2 A_{3} = 2 \cdot 9 \pi r^{2} = 18 \pi r^{2}$

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.

Report an Error

Example Question #7 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

Which is the greater quantity?

(a) Twice the area of inner gray ring

(b) The area of the white ring

Possible Answers:

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(a) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; 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 $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$

The white ring has as its area the difference of the areas of the second-largest and third-largest circles:

$A_{3} - A_{2} = 9 \pi - 4 \pi = 5 \pi$

The inner gray ring has as its area the difference of the areas of the third-largest and smallest circles:

$A_{2} - A_{1} = 4 \pi - \pi = 3 \pi$ .

Twice this is $2 \cdot 3 \pi = 6 \pi$ , which is greater than the area of the white ring.

Report an Error

Example Question #2 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

Which is the greater quantity?

(a) Six times the area of the white circle

(b) The area of the outer ring

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(b) is the greater quantity

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; 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 $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$

The outer gray ring is the region between the largest and second-largest circles, and has area

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$

Six times the area of the white (inner) circle is $6 A_{1} = 6 \pi$ , which is less than the area of the outer ring, $7 \pi$ .

Report an Error

Example Question #151 : Geometry

A giant clock has a minute hand five yards in length. Since noon, the tip of the minute hand has traveled $200 \pi$ feet. Which is the greater quantity?

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

(B) The amount of time until midnight

Possible Answers:

(A) and (B) are equal

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

(A) is greater

(B) is greater

Correct answer:

(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 $2 \pi$ times this, or

$C = 2 \pi \cdot 15 = 30 \pi$ feet.

In one six-hour period, the minute hand revolves six times, so its tip travles six times the circumference, or

$30 \pi \times 6 = 180 \pi$

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.

Report an Error

Example Question #152 : Geometry

Compare the two quantities:

Quantity A: The area of a circle with radius

Quantity B: The circumference of a circle with radius 8.1

Possible Answers:

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.

Correct answer:

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:

$A=\pi * r^2$

$C=2*\pi*r$

For our two quantities, we have:

Quantity A: $\pi * 5^2 = 25\pi$

Quantity B: $2 * \pi * 8.1 = 16.2\pi$

Therefore, quantity A is greater.

Report an Error

Example Question #161 : Geometry

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.

Possible Answers:

3 feet 10 inches

1 foot, 11 inches

2 feet

4 feet 2 inches

2 feet, 1 inch

Correct answer:

1 foot, 11 inches

Explanation:

The circumference of a circle can be determined by multiplying its radius by $2 \pi$ , so the circumferences of the two smallest circles are

$C_{1} = 2 \cdot 2 \pi = 4 \pi$

and

$C_{2} = 2 \cdot 5 \pi = 10 \pi$

The circumferences form an arithmetic sequence with common difference

$C_{2}- C_{1}= 10 \pi - 4 \pi = 6 \pi$

The circumference of a circle can therefore be found using the formula

$C_{n} = C_{1} + (n-1)d$

where $C_{1} = 4 \pi$ and $d = 6 \pi$ ; we are looking for that of the n = 8 th smallest circle, so

$C_{8} = 4 \pi + (8-1)6 \pi$

$= 4 \pi + 7 \cdot 6 \pi$

$= 4 \pi + 42 \pi$

$= 46 \pi$

Since the radius of a circle is the circumference of the circle divided by $2 \pi$ , the radius of this eighth circle is

$r = \frac{46 \pi}{2 \pi} = 23$ inches, or 1 foot 11 inches.

Report an Error

Example Question #162 : Geometry

Track

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?

Possible Answers:

Correct answer:

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

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 400 = 628$ feet.

The distance around the track is about

400 + 628 + 400 + 400 = 1,828 feet.

Diane wants to run two miles, or

$5,280 \times 2 = 10, 560$ feet.

She will make about

$10,560 \div 1,828 \approx 5.78$

circuits around the track.

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

$5 \times 1,828 = 9,140$ feet,

so she will have

10, 560 - 9.140=1,420 feet to go.

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

1,420-800 = 620 feet.

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

The correct response is Point B.

Report an Error

Example Question #5 : How To Find Circumference

Track

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)

Possible Answers:

(b) is greater

(a) is greater

(a) and (b) are equal

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

Correct answer:

(b) is greater

Explanation:

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 $1\frac{1}{5} = \frac{6}{5}$ times. Mike runs around the track clockwise from Point A to Point E, in the same time, meaning he runs around the track $\frac{4}{5}$ times.

Therefore, Evan's speed is $\frac{ \frac{6}{5}}{\frac{4}{5}} = \frac{3}{2}$ times Mike's speed. As a result, Twice Mike's speed would be greater than Evan's speed, making (b) the greater.

Report an Error

View Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Tutors

Robert
Certified Tutor

University of North Carolina at Chapel Hill, Bachelor in Arts, English. North Carolina Central University, Master of Arts, En...

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Denver, Calculus Tutors in Miami, SAT Tutors in Atlanta, GMAT Tutors in San Francisco-Bay Area, GRE Tutors in New York City, English Tutors in New York City, SSAT Tutors in Boston, Chemistry Tutors in Boston, SAT Tutors in Dallas Fort Worth, Statistics Tutors in Denver

Popular Courses & Classes

GRE Courses & Classes in Los Angeles, ACT Courses & Classes in Philadelphia, MCAT Courses & Classes in Phoenix, MCAT Courses & Classes in Denver, ACT Courses & Classes in New York City, ACT Courses & Classes in Atlanta, LSAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Atlanta, Spanish Courses & Classes in Chicago, ISEE Courses & Classes in Dallas Fort Worth

Popular Test Prep

GRE Test Prep in Denver, SAT Test Prep in Houston, ACT Test Prep in Atlanta, MCAT Test Prep in Philadelphia, GRE Test Prep in Los Angeles, MCAT Test Prep in Denver, ISEE Test Prep in Dallas Fort Worth, ISEE Test Prep in Seattle, ACT Test Prep in Boston, ACT Test Prep in Houston

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Quantitative : Circles

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

Example Question #6 : How To Find The Area Of A Circle

Example Question #7 : How To Find The Area Of A Circle

Example Question #2 : How To Find The Area Of A Circle

Example Question #151 : Geometry

Example Question #152 : Geometry

Example Question #161 : Geometry

Example Question #162 : Geometry

Example Question #5 : How To Find Circumference

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: