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If each of the 4 circles above has radius 1,

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If each of the 4 circles above has radius 1, [#permalink] New post 29 May 2018, 05:25
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If each of the 4 circles above has radius 1, what is the area of the shaded region?
A) 1 – π/4
B) 2 – π/4
C) 2 – π/2
D) 4 – π
E) 4 – 2π
[Reveal] Spoiler: OA

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Re: If each of the 4 circles above has radius 1, [#permalink] New post 29 May 2018, 23:38
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In these four identical circles let us assume the center of the circle from left uppermost to left bottom most are a,b,c and d respectively

If we connect the center of the 4 circles we will get a square. Each side of the square will be of length 2

Now, the area of the square would be \(2 *2 = 4\)
We are required to find the area of the shaded region hence we should reduce the portion of the circle enclosed within the square that is not shaded.
Since the side of the square are at 90 degrees to each other we know that there are 4 right triangles within the square enclosed in the circle.

For this we should find the area of 4 sectors that are not shaded.
The area of each sector would be equal to the other three sector.

area of 1 sector = \(\frac{90}{360}\) * ∏ * \(r^2 = \frac{1}{4} *\) ∏ as radius = 1
for 4 sectors area would be \(\frac{1}{4} *\) ∏ \(* 4\) = ∏

Therefore, area of the shaded region = 4 - ∏
option D
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Re: If each of the 4 circles above has radius 1, [#permalink] New post 30 May 2018, 16:27
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Outstanding explanation.

It should be put as sticky "answer of the month".

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Re: If each of the 4 circles above has radius 1, [#permalink] New post 31 May 2018, 05:35
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GreenlightTestPrep wrote:
Image

If each of the 4 circles above has radius 1, what is the area of the shaded region?
A) 1 – π/4
B) 2 – π/4
C) 2 – π/2
D) 4 – π
E) 4 – 2π


Here's a different approach:

Let's place a square around just one of the circles
Image


Notice that the shaded area IN the square represents 1/4 of the TOTAL shaded area in the diagram.
Image


This means that the shaded area in THIS diagram will be the same as the TOTAL shaded area in the ORIGINAL diagram.
Image


So, let's determine the area of the shaded area in the diagram below.
Image

Each side of the square has length 2, so the area of the SQUARE = (2)(2) = 4

The circle has radius 1.
So, the area of the CIRCLE = π(radius)² = π(1)² = π

So, the area of the SHADED region = 4 - π

Answer: D

Cheers,
Brent
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Re: If each of the 4 circles above has radius 1, [#permalink] New post 04 Jun 2018, 17:42
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GreenlightTestPrep wrote:
GreenlightTestPrep wrote:
Image

If each of the 4 circles above has radius 1, what is the area of the shaded region?
A) 1 – π/4
B) 2 – π/4
C) 2 – π/2
D) 4 – π
E) 4 – 2π


Here's a different approach:

Let's place a square around just one of the circles
Image


Notice that the shaded area IN the square represents 1/4 of the TOTAL shaded area in the diagram.
Image


This means that the shaded area in THIS diagram will be the same as the TOTAL shaded area in the ORIGINAL diagram.
Image


So, let's determine the area of the shaded area in the diagram below.
Image

Each side of the square has length 2, so the area of the SQUARE = (2)(2) = 4

The circle has radius 1.
So, the area of the CIRCLE = π(radius)² = π(1)² = π

So, the area of the SHADED region = 4 - π

Answer: D

Cheers,
Brent

This approach would save some time

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Re: If each of the 4 circles above has radius 1,   [#permalink] 04 Jun 2018, 17:42
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