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# Three identical circles are drawn in a rectangle as shown above. If th

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Math Expert
Joined: 02 Sep 2009
Posts: 65841
Three identical circles are drawn in a rectangle as shown above. If th  [#permalink]

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19 Jun 2020, 03:07
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Difficulty:

25% (medium)

Question Stats:

88% (01:59) correct 13% (04:09) wrong based on 24 sessions

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Three identical circles are drawn in a rectangle as shown above. If the radius of each circle is 1, what is the area of the rectangle?

A. $$8+\sqrt{3}$$

B. $$8+2\sqrt{3}$$

C. $$8+3\sqrt{3}$$

D. $$8+4\sqrt{3}$$

E. $$8+5\sqrt{3}$$

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Re: Three identical circles are drawn in a rectangle as shown above. If th  [#permalink]

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19 Jun 2020, 03:17
2
Resulting an equilateral triangle of side 2
The height of equilateral triangle = (Side/2)*√3
Height = 2/2*√3
Height = √3
=> 2+√3
=> 4*1 => 4

Area of rectangle = 4*(2+√3)
=> 8+4√3

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Re: Three identical circles are drawn in a rectangle as shown above. If th  [#permalink]

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19 Jun 2020, 03:29
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Attachment:

2020-06-19_1504.png [ 21.34 KiB | Viewed 358 times ]

L =1+1+1+1 = 4

$$AD = \sqrt{2^2-1^2}$$ = √3

B =√3+2

L*B = 4*(√3+2) = 4√3+8
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Joined: 04 Mar 2020
Posts: 76
Re: Three identical circles are drawn in a rectangle as shown above. If th  [#permalink]

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19 Jun 2020, 03:46
1
Option D

Let's denote the center of each circle as A, B, and C starting from top left and moving clockwise.
Draw a straight line passing through centers A and B. This is gives us the one side of the rectangle. which is nothing but 4 times the radius. lets call it S1= 4.

Notice that the centers A B and C form an equilateral triangle with side 2.
drop a perpendicular from C on line AB. Lets say it meets at point O

for the triangle AOC
AC = 2; AO=1 and OC = $$\sqrt{3}$$ (using Pythagorean theorem)

using these we can find the other side on rectangle $$S2= \sqrt{3} + 1 + 1$$

S2 = 2+ $$\sqrt{3}$$

Area = S1*S2

$$4*(2+\sqrt{3})$$
$$8+4\sqrt{3}$$
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Re: Three identical circles are drawn in a rectangle as shown above. If th  [#permalink]

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20 Jun 2020, 10:45
1
Kindly see the attachment.

width = b
The length is calculated in the right side.
D
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Re: Three identical circles are drawn in a rectangle as shown above. If th   [#permalink] 20 Jun 2020, 10:45