Last visit was: 23 Apr 2026, 23:55 It is currently 23 Apr 2026, 23:55
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
705-805 (Hard)|   Geometry|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,802
Own Kudos:
810,902
 [27]
Given Kudos: 105,868
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,802
Kudos: 810,902
 [27]
Six congruent circles are packed into an equilateral triangle so that 08 Mar 2018, 21:17
2
Kudos
Add Kudos
25
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

85% (hard)

Question Stats:

47% (01:58) correct 53% (01:50) wrong based on 310 sessions

History

Date
Time
Result
Not Attempted Yet


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
six_circles_packed_1.png
six_circles_packed_1.png [ 16.87 KiB | Viewed 12157 times ]
ShowHide Answer
Official Answer
D
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,802
Own Kudos:
810,902
 [6]
Given Kudos: 105,868
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,802
Kudos: 810,902
 [6]
Six congruent circles are packed into an equilateral triangle so that 08 Mar 2018, 21:22
Kudos
Add Kudos
6
Bookmarks
Bookmark this Post
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
six_circles_packed_1.png
Show more

23. Geometry




24. Coordinate Geometry




25. Triangles




26. Polygons




27. Circles




28. Rectangular Solids and Cylinders




29. Graphs and Illustrations



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread
General Discussion
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 23 Apr 2026
Posts: 6,976
Own Kudos:
16,909
 [3]
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
My Reviews
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,976
Kudos: 16,909
 [3]
Six congruent circles are packed into an equilateral triangle so that 09 Mar 2018, 01:07
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
The attachment six_circles_packed_1.png is no longer available
Show more

Question: Area of uncovered part of triangle = Area of triangle - Area of six circles = ?

Statement 1: The radius of each circle is 2
Using this information the side of the equilateral triangle can be calculated (using 30-60-90 property) as mentioned in attachment, Hence
SUFFICIENT

Statement 2: he area of the triangle is \(48+28\sqrt{3}\)
sing this information the side of the equilateral triangle can be calculated as mentioned in attachment, Hence
SUFFICIENT

Answer: option D
Attachments

File comment: www.GMATinsight.com
Screen Shot 2018-03-09 at 1.40.56 PM.png
Screen Shot 2018-03-09 at 1.40.56 PM.png [ 204.67 KiB | Viewed 10603 times ]

User avatar
GMATBusters
User avatar
GMAT Tutor
Joined: 27 Oct 2017
Last visit: 23 Apr 2026
Posts: 1,922
Own Kudos:
6,856
 [2]
Given Kudos: 241
WE:General Management (Education)
Expert
Expert reply
Posts: 1,922
Kudos: 6,856
 [2]
Six congruent circles are packed into an equilateral triangle so that 10 Mar 2018, 20:42
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
There is a simpler approach to it, by trying to draw it on paper.
see the Sketch attached.
Attachment:
WhatsApp Image 2018-03-11 at 09.09.28.jpeg
WhatsApp Image 2018-03-11 at 09.09.28.jpeg [ 99.73 KiB | Viewed 10478 times ]

GMATinsight
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
The attachment six_circles_packed_1.png is no longer available

Question: Area of uncovered part of triangle = Area of triangle - Area of six circles = ?

Statement 1: The radius of each circle is 2
Using this information the side of the equilateral triangle can be calculated (using 30-60-90 property) as mentioned in attachment, Hence
SUFFICIENT

Statement 2: he area of the triangle is \(48+28\sqrt{3}\)
sing this information the side of the equilateral triangle can be calculated as mentioned in attachment, Hence
SUFFICIENT

Answer: option D
Show more
User avatar
fskilnik
User avatar
Joined: 12 Oct 2010
Last visit: 03 Jan 2025
Posts: 883
Own Kudos:
1,882
Given Kudos: 57
Status:GMATH founder
Expert
Expert reply
Posts: 883
Kudos: 1,882
Six congruent circles are packed into an equilateral triangle so that 26 Oct 2018, 08:44
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)
Show more
\(?\,\,\, = \,\,\,{S_\Delta } - 6 \cdot {S_ \circ }\)

The GMATH widget (a.k.a "GW") is defined in the figure below. It is formed combining an equilateral triangle (side length 4r) and three "legs" (length r each), where r is any positive number.



With the GW in mind, the answer is (D) immediately:

(1) The value of r is given, hence the corresponding GW is unique, hence the triangle that circumscribes the GW is unique (dotted triangle in the figure shown in the bottom-left). Our FOCUS is unique.

(2) The equilateral triangle is given, hence the corresponding inscribed GW is unique (dotted GW shown in the bottom-right), hence r is unique. Our FOCUS is unique.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
User avatar
CEdward
User avatar
Joined: 11 Aug 2020
Last visit: 14 Apr 2022
Posts: 1,162
Own Kudos:
289
Given Kudos: 332
Posts: 1,162
Kudos: 289
Six congruent circles are packed into an equilateral triangle so that 11 Mar 2021, 22:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GMATinsight
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
six_circles_packed_1.png

Question: Area of uncovered part of triangle = Area of triangle - Area of six circles = ?

Statement 1: The radius of each circle is 2
Using this information the side of the equilateral triangle can be calculated (using 30-60-90 property) as mentioned in attachment, Hence
SUFFICIENT

Statement 2: he area of the triangle is \(48+28\sqrt{3}\)
sing this information the side of the equilateral triangle can be calculated as mentioned in attachment, Hence
SUFFICIENT

Answer: option D
Show more

Can you explain statement 2? How does knowing the side length allow us to get the area of the circle?

Re-did this question.

To clarify, both statements are sufficient. The key is knowing that a 30-60-90 triangle forms at each corner of the equilateral triangle. If we can get the radius of the circle, then we can solve.

For statement 2, we can get the side length from knowing the area of the equilateral triangle. That side length in turn is equal to (4 x radius) + (2 x radius√3).

D.
User avatar
HaSmamit
User avatar
Joined: 14 Apr 2022
Last visit: 25 May 2025
Posts: 32
Own Kudos:
16
 [1]
Given Kudos: 69
Posts: 32
Kudos: 16
 [1]
Six congruent circles are packed into an equilateral triangle so that 01 Nov 2023, 07:43
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
CEdward
GMATinsight
Bunuel


Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

(1) The radius of each circle is 2
(2) The area of the triangle is \(48+28\sqrt{3}\)


Show Spoiler
Attachment:
six_circles_packed_1.png

Question: Area of uncovered part of triangle = Area of triangle - Area of six circles = ?

Statement 1: The radius of each circle is 2
Using this information the side of the equilateral triangle can be calculated (using 30-60-90 property) as mentioned in attachment, Hence
SUFFICIENT

Statement 2: he area of the triangle is \(48+28\sqrt{3}\)
sing this information the side of the equilateral triangle can be calculated as mentioned in attachment, Hence
SUFFICIENT

Answer: option D

Can you explain statement 2? How does knowing the side length allow us to get the area of the circle?

Re-did this question.

To clarify, both statements are sufficient. The key is knowing that a 30-60-90 triangle forms at each corner of the equilateral triangle. If we can get the radius of the circle, then we can solve.

For statement 2, we can get the side length from knowing the area of the equilateral triangle. That side length in turn is equal to (4 x radius) + (2 x radius√3).

D.
Show more

Hi CEdward
How can I know that a 30-60-90 triangle forms at each corner of the equilateral triangle?
Moderators:
Bunuel
Math Expert
109802 posts
kingbucky
498 posts
Gmat860sanskar
212 posts