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# Square ABCD is perfectly inscribed in the circle pictured above. If mi

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Math Expert
Joined: 02 Sep 2009
Posts: 55614
Square ABCD is perfectly inscribed in the circle pictured above. If mi  [#permalink]

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06 Jun 2017, 11:14
00:00

Difficulty:

55% (hard)

Question Stats:

70% (02:45) correct 30% (02:41) wrong based on 47 sessions

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Square ABCD is perfectly inscribed in the circle pictured above. If minor arc AD measures 2π, what is the approximate area of the shaded region?

A. 110
B. 72
C. 36
D. 18
E. 9

Attachment:

Inscribed.png [ 10.76 KiB | Viewed 1220 times ]

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Joined: 24 Apr 2016
Posts: 328
Re: Square ABCD is perfectly inscribed in the circle pictured above. If mi  [#permalink]

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06 Jun 2017, 11:41
As Square ABCD is perfectly inscribed in the circle, all the four arcs will be equal

So the total length of all 4 arcs = circumference of the circle = 4 x 2π = 8π

Radius of the circle =$$\frac{8π}{2π}$$= 4

Area of the circle = π X 16 = 16π

Diameter of the Circle = Diagonal of the Square

If each side of the square is a, then 2$$a^2$$ = $$8^2$$==> a = $$\frac{8}{\sqrt{2}}$$

Area of Square = $$(8/\sqrt{2})^2$$ = 32

Area of Shaded Region = Area of Circle - Area of Square = 16π - 32 = 18.2

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Re: Square ABCD is perfectly inscribed in the circle pictured above. If mi  [#permalink]

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25 May 2019, 05:50
Hi, i just wanted to understand why all the arcs are equal? I read the similar concept when the arcs of a circle were circumscribed by an equilateral triangle. is there a concept i am missing out on.
Senior Manager
Joined: 19 Oct 2018
Posts: 470
Location: India
Re: Square ABCD is perfectly inscribed in the circle pictured above. If mi  [#permalink]

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25 May 2019, 08:42
Each arc is making same angle (90*) at the center.
abhishek31 wrote:
Hi, i just wanted to understand why all the arcs are equal? I read the similar concept when the arcs of a circle were circumscribed by an equilateral triangle. is there a concept i am missing out on.
Re: Square ABCD is perfectly inscribed in the circle pictured above. If mi   [#permalink] 25 May 2019, 08:42
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