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# In the figure above, a circle with center O is inscribed in square ABC

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Math Expert
Joined: 02 Sep 2009
Posts: 50001
In the figure above, a circle with center O is inscribed in square ABC  [#permalink]

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23 Nov 2017, 01:24
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Difficulty:

25% (medium)

Question Stats:

92% (01:47) correct 8% (01:34) wrong based on 19 sessions

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In the figure above, a circle with center O is inscribed in square ABCD. What is the total area of the shaded regions?

(A) 4 – π/2
(B) 8 – 2π
(C) 8 – 3π/2
(D) 16 – 4π
(E) 16 – 2π

Attachment:

2017-11-23_1215_002.png [ 7.13 KiB | Viewed 1068 times ]

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In the figure above, a circle with center O is inscribed in square ABC  [#permalink]

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23 Nov 2017, 12:37
Bunuel wrote:

In the figure above, a circle with center O is inscribed in square ABCD. What is the total area of the shaded regions?

(A) 4 – π/2
(B) 8 – 2π
(C) 8 – 3π/2
(D) 16 – 4π
(E) 16 – 2π

Attachment:
2017-11-23_1215_002.png

Area of square minus circle yields 4 regions' area

(Square area) - (Circle area) = area of all 4 equal small corner regions

Square side length = $$4$$
Area of square = $$s^2 = 16$$

Square's side length = circle's diameter, $$d$$ and $$d = 2r$$
$$4 = 2r$$
$$r = 2$$
Circle's area: $$πr^2 = 4π$$

(Square area) - (circle area)= Area of 4 regions= $$16 - 4π$$

2 of 4 regions are shaded
So if (Square area) - (circle area) = 4 regions

Then (Square area) - (circle area) divided by 2 = area of 2 regions

$$\frac{16 - 4π}{2}$$ = 2 shaded regions

Shaded regions' area = $$8 - 2π$$

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Joined: 11 Nov 2017
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Re: In the figure above, a circle with center O is inscribed in square ABC  [#permalink]

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23 Nov 2017, 13:34
In the figure above, a circle with center O is inscribed in square ABCD. What is the total area of the shaded regions?

(A) 4 – π/2
(B) 8 – 2π
(C) 8 – 3π/2
(D) 16 – 4π
(E) 16 – 2π

Attachment:
2017-11-23_1215_002.png
[/quote]

Well, this is an easy one.

We basically need to find the are not covered by the circle, within the square and then divide than area in half.
Area of square = 4^2=16
Area of circle = (2^2)pi [diameter = side of square=4. hence, radius=2]
Area of the shaded region = (16-4pi)/2= 8-2pi

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Re: In the figure above, a circle with center O is inscribed in square ABC &nbs [#permalink] 23 Nov 2017, 13:34
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