Last visit was: 24 Apr 2026, 00:31 It is currently 24 Apr 2026, 00:31
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
805+ (Hard)|   Geometry|      
User avatar
AliciaSierra
User avatar
Joined: 17 Mar 2014
Last visit: 14 Jun 2024
Posts: 736
Own Kudos:
648
 [20]
Given Kudos: 1,350
Products:
My Reviews
Posts: 736
Kudos: 648
 [20]
If a regular octagon is inscribed in a square 28 Jul 2017, 12:31
2
Kudos
Add Kudos
18
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

36% (02:51) correct 64% (02:27) wrong based on 126 sessions

History

Date
Time
Result
Not Attempted Yet
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
 [7]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
 [7]
If a regular octagon is inscribed in a square 29 Jul 2017, 03:03
3
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7
Show more


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A
Attachments

octagon.png
octagon.png [ 8.31 KiB | Viewed 16790 times ]

General Discussion
User avatar
gracie
User avatar
Joined: 07 Dec 2014
Last visit: 11 Oct 2020
Posts: 1,028
Own Kudos:
2,022
 [1]
Given Kudos: 27
Posts: 1,028
Kudos: 2,022
 [1]
If a regular octagon is inscribed in a square 29 Jul 2017, 21:09
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7
Show more

let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E
Attachments

inscribed octagon.png
inscribed octagon.png [ 8.13 KiB | Viewed 16464 times ]

User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
 [1]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
 [1]
If a regular octagon is inscribed in a square 29 Jul 2017, 21:31
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
gracie
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7

let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E
Show more


Hi...

A REGULAR octagon is an octagon with EQUAL sides..

but in your sketch, that is not the case.
4 sides are equal to SIDE of square and other 4 sides are equal to DIAGONAL of square..
that is why the answer is wrong.
User avatar
gracie
User avatar
Joined: 07 Dec 2014
Last visit: 11 Oct 2020
Posts: 1,028
Own Kudos:
2,022
Given Kudos: 27
Posts: 1,028
Kudos: 2,022
If a regular octagon is inscribed in a square 29 Jul 2017, 22:50
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
gracie
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7

let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E


Hi...

A REGULAR octagon is an octagon with EQUAL sides..

but in your sketch, that is not the case.
4 sides are equal to SIDE of square and other 4 sides are equal to DIAGONAL of square..
that is why the answer is wrong.
Show more

hi chetan2u,
thank you for pointing out my error.
gracie
avatar
royjoseph15
avatar
Joined: 18 May 2019
Last visit: 15 Jan 2020
Posts: 1
Given Kudos: 14
Posts: 1
Kudos: 0
If a regular octagon is inscribed in a square 04 Nov 2019, 22:50
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A
Show more

Can you explain - how you got the Side of the square after assuming that the side of the Octagon is a√2?

Also, are all sides of Octagons always equal?
User avatar
harsh8686
User avatar
Joined: 19 Feb 2018
Last visit: 23 Feb 2024
Posts: 97
Own Kudos:
166
Given Kudos: 229
GMAT 1: 530 Q48 V15
GMAT 2: 640 Q48 V30
GMAT 3: 590 Q46 V25
GMAT 4: 600 Q48 V25 (Online)
GMAT 5: 710 Q49 V37
GMAT 5: 710 Q49 V37
Posts: 97
Kudos: 166
If a regular octagon is inscribed in a square 22 May 2021, 23:25
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A
Show more

Can you please explain the highlighted part? I think it is wrong. The area of square should be \(a^2(2+\sqrt{2})^2\)
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
If a regular octagon is inscribed in a square 23 May 2021, 00:08
Kudos
Add Kudos
Bookmarks
Bookmark this Post
harsh8686
chetan2u
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A

Can you please explain the highlighted part? I think it is wrong. The area of square should be \(a^2(2+\sqrt{2})^2\)
Show more

Both are the same

\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)
User avatar
harsh8686
User avatar
Joined: 19 Feb 2018
Last visit: 23 Feb 2024
Posts: 97
Own Kudos:
166
 [1]
Given Kudos: 229
GMAT 1: 530 Q48 V15
GMAT 2: 640 Q48 V30
GMAT 3: 590 Q46 V25
GMAT 4: 600 Q48 V25 (Online)
GMAT 5: 710 Q49 V37
GMAT 5: 710 Q49 V37
Posts: 97
Kudos: 166
 [1]
If a regular octagon is inscribed in a square 28 May 2021, 09:10
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
chetan2u


Both are the same

\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)
Show more

But in the solution, you have written
\((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
If a regular octagon is inscribed in a square 28 May 2021, 09:41
Kudos
Add Kudos
Bookmarks
Bookmark this Post
harsh8686
chetan2u


Both are the same

\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)

But in the solution, you have written
\((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
Show more


\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)

Take out \(\sqrt{2}\) to get what you are asking about.

\(a^2(\sqrt{2}*\sqrt{2}+\sqrt{2})^2\)
\(a^2(\sqrt{2})^2(\sqrt{2}+1)^2\)
\(2a^2(\sqrt{2}+1)^2\)
User avatar
sid070999
User avatar
Joined: 02 Jan 2021
Last visit: 16 Sep 2025
Posts: 6
Own Kudos:
1
Given Kudos: 3
Posts: 6
Kudos: 1
If a regular octagon is inscribed in a square 18 Jan 2024, 07:29
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A
Show more

how you considered those triangles formed to be right isosceles triangle? please explain
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
 [1]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
 [1]
If a regular octagon is inscribed in a square 18 Jan 2024, 08:04
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sid070999
chetan2u
ammuseeru
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
A

how you considered those triangles formed to be right isosceles triangle? please explain
Show more


Hi

Side of Regular octagon will be in the middle of the side of square as also shown in the sketch in initial post. let the portion be 'a' on either end.
Each side of the square will be a+(side of octagon)+a.

Thus, each vertices of square will have small portions 'a' as the two side and third side will be the hypotenuse, which is nothing but the side of Octagon.

As each vertices of a square makes a right angle, the corner triangle formed will be a:a:a\(\sqrt{2}\), an isosceles right angled triangle.
User avatar
sid070999
User avatar
Joined: 02 Jan 2021
Last visit: 16 Sep 2025
Posts: 6
Own Kudos:
1
Given Kudos: 3
Posts: 6
Kudos: 1
If a regular octagon is inscribed in a square 18 Jan 2024, 11:03
Kudos
Add Kudos
Bookmarks
Bookmark this Post
But every octagon side trisects every square sides right? so if one of the side of octagon is a square root 2 so the other two portions as shown in the sketch cant they be a square root 2?
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
45,006
 [1]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,006
 [1]
If a regular octagon is inscribed in a square 18 Jan 2024, 11:28
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sid070999
But every octagon side trisects every square sides right? so if one of the side of octagon is a square root 2 so the other two portions as shown in the sketch cant they be a square root 2?
Show more


No, it does not. Although a sketch would be easier to explain, but I will try through words.

Let us assume that it does trisect.
so you have each side of square is 3x, and is trisected as x+x+x. Also, every side of octagon too is x.
If this is true, take the triangles on any corner of square. The triangle will have two sides x each on its neighboring sides and the third side will also be x, given by the octagonal side. So, the triangle is x-x-x or an equilateral triangle. This means all angle of the triangle including the corner angle is 60. But the square has to have all angles as 90.
Thus, our assumption that each side of square is trisected is wrong.

Ill add a video or a sketch if this doesnt help.
User avatar
samarpan.g28
User avatar
Joined: 08 Dec 2023
Last visit: 18 Feb 2026
Posts: 315
Own Kudos:
136
Given Kudos: 1,236
Location: India
Concentration: General Management, Human Resources
Schools: IIMA '26 Schulich Sept '26 Montreal '26 HEC Sept '26
GPA: 8.88
WE:Engineering (Technology)
Schools: IIMA '26 Schulich Sept '26 Montreal '26 HEC Sept '26
Posts: 315
Kudos: 136
If a regular octagon is inscribed in a square 18 Jan 2024, 13:38
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sid070999
But every octagon side trisects every square sides right? so if one of the side of octagon is a square root 2 so the other two portions as shown in the sketch cant they be a square root 2?
Show more

I hope this helps. Please remember, we have to keep the sides of the octagon the same as I have considered x here, but the sides of squares will be obtained based on the sides of the octagon.
Attachments

2d0b7f6a-255a-41b9-a7ab-615c549b4efc.jpeg
2d0b7f6a-255a-41b9-a7ab-615c549b4efc.jpeg [ 132.83 KiB | Viewed 3818 times ]

Moderators:
Bunuel
Math Expert
109802 posts
Krunaal
Tuck School Moderator
853 posts