GMAT Math : DSQ: Calculating the area of a sector

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Dsq: Calculating The Area Of A Sector

Weird

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

2 r + 2 (2r) = 60

Either statement alone allows us to find the radius and, consequently, the area of the white region.

Report an Error

Example Question #2 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The circle has circumference $40 \pi$ .

Statement 2: $m \angle ACB = 30 ^{\circ }$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

Report an Error

Example Question #3 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or 5:1 .

Report an Error

Example Question #4 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: AD = 100

Statement 2: $m \angle ACB = 30 ^{\circ }$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or 2:1 .

Report an Error

Example Question #5 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

Report an Error

Example Question #6 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

r N= 1,440

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

r = 12 ; the second statement can be solved for :

12 N= 1,440

N = 120 .

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since r = 12 , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

Report an Error

Example Question #7 : Dsq: Calculating The Area Of A Sector

What is the area of a $60^{\circ}$ sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is $8 \pi$ inches.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

24=r

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Miami, ISEE Tutors in San Francisco-Bay Area, Chemistry Tutors in Atlanta, Reading Tutors in Philadelphia, Biology Tutors in San Francisco-Bay Area, Algebra Tutors in Chicago, SSAT Tutors in Dallas Fort Worth, Statistics Tutors in San Francisco-Bay Area, English Tutors in Houston, Spanish Tutors in Washington DC

Popular Courses & Classes

ISEE Courses & Classes in Houston, SAT Courses & Classes in Miami, GMAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Boston, Spanish Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Seattle, ISEE Courses & Classes in San Diego, MCAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Atlanta, GMAT Courses & Classes in Seattle

Popular Test Prep

LSAT Test Prep in Philadelphia, LSAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Dallas Fort Worth, GMAT Test Prep in Miami, MCAT Test Prep in Seattle, GRE Test Prep in San Diego, GMAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Houston, GRE Test Prep in Denver, SAT Test Prep in Boston

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)

Email address:
Your name:
Feedback:

GMAT Math : DSQ: Calculating the area of a sector

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Calculating The Area Of A Sector

Example Question #2 : Dsq: Calculating The Area Of A Sector

Example Question #3 : Dsq: Calculating The Area Of A Sector

Example Question #4 : Dsq: Calculating The Area Of A Sector

Example Question #5 : Dsq: Calculating The Area Of A Sector

Example Question #6 : Dsq: Calculating The Area Of A Sector

Example Question #7 : Dsq: Calculating The Area Of A Sector

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details