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# M24-05

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Math Expert
Joined: 02 Sep 2009
Posts: 58453

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16 Sep 2014, 01:20
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85% (hard)

Question Stats:

45% (01:11) correct 55% (01:37) wrong based on 177 sessions

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A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

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Math Expert
Joined: 02 Sep 2009
Posts: 58453

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16 Sep 2014, 01:20
5
2
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

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19 Mar 2015, 06:06
I think this question is good and helpful.
Hi, could you please explain why the area of the inscribed square in a circle with the diameter of 2 is 2?
Math Expert
Joined: 02 Sep 2009
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19 Mar 2015, 06:13
IonutCZ wrote:
I think this question is good and helpful.
Hi, could you please explain why the area of the inscribed square in a circle with the diameter of 2 is 2?

The area of a square = diagonal^2/2 = 2^2/2 = 2.
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26 Sep 2016, 01:08
After reading this explanation, I came up with this doubt: If we draw the diagonals of the rectangle (the blue region), each of the diagonals would pass through the center which- would be same as the diameter of the circle. Right or wrong? I am not able to prove this wrong.
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26 Sep 2016, 02:34
1
La1yaMalhotra wrote:
After reading this explanation, I came up with this doubt: If we draw the diagonals of the rectangle (the blue region), each of the diagonals would pass through the center which- would be same as the diameter of the circle. Right or wrong? I am not able to prove this wrong.

Right. A rectangle has right angles. A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle). The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled (right angel being the angle opposite the diameter/hypotenuse)
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26 Sep 2016, 05:51
Then should the area of the rectangle be 1/2*d1*d2= 1/2*2*2=2?. I think I am getting this wrong, but where I don't know.
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Posts: 58453

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27 Sep 2016, 01:37
La1yaMalhotra wrote:
Then should the area of the rectangle be 1/2*d1*d2= 1/2*2*2=2?. I think I am getting this wrong, but where I don't know.

If the rectangle is a square then the area would be 2 (as given in the solution). Don't understand what you derive from this and what's your doubt.
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05 Apr 2017, 10:55
This is a great question. Very enlightening. There is a question about a small ''technicallity'' though.

The question stem asks about a ''rectangle''. Since the area of 2 is only achieved throught a square isn't it provoking a little doubt that you could marginally discard the area =2 because ''technically'' the questions asks about rectangle ?

I hope my question is not awkwardly written
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05 Apr 2017, 21:46
alexlovesgmat wrote:
This is a great question. Very enlightening. There is a question about a small ''technicallity'' though.

The question stem asks about a ''rectangle''. Since the area of 2 is only achieved throught a square isn't it provoking a little doubt that you could marginally discard the area =2 because ''technically'' the questions asks about rectangle ?

I hope my question is not awkwardly written

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All squares are rectangles.
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26 Oct 2017, 12:42
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!
Math Expert
Joined: 02 Sep 2009
Posts: 58453

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26 Oct 2017, 21:41
manishtank1988 wrote:
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!

Not sure what you are doing there. If a square is inscribed in a circle with the diameter of 2, then the area of the square would simply be diagonal^2/2 = 4/2 = 2 (because diameter=diagonal).
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27 Oct 2017, 09:14
Bunuel wrote:
manishtank1988 wrote:
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!

Not sure what you are doing there. If a square is inscribed in a circle with the diameter of 2, then the area of the square would simply be diagonal^2/2 = 4/2 = 2 (because diameter=diagonal).

Hello Bunuel
I got my mistake - we are talking about largest rectangle (square) within a circle - thus area of square - a^2 => 2.
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12 Apr 2018, 07:32
+1 for option D. The key take away is that one shouldn't fail to consider boundary cases while evaluating options. Here if you overlook the fact that all squares are rectangle , you are bound to get it wrong !!! Great question
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23 Aug 2018, 00:20
Can we say that because the rectangle is inside the circle, the area of the rectangle must be <= that of circle.
the area of the circle is 3.14, therefore the area of the rectangle must be <=3.14?
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Posts: 58453

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23 Aug 2018, 00:28
soleimanian wrote:
Can we say that because the rectangle is inside the circle, the area of the rectangle must be <= that of circle.
the area of the circle is 3.14, therefore the area of the rectangle must be <=3.14?

Yes, the are of any inscribed figure in a circle is LESS (strictly less, not less than or equal) to that of the circle. This is mentioned in the solution too.
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Re: M24-05   [#permalink] 23 Aug 2018, 00:28
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# M24-05

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