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# The corners of an equilateral triangle are rounded to form semi-circle

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Math Expert
Joined: 02 Sep 2009
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The corners of an equilateral triangle are rounded to form semi-circle  [#permalink]

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14 Sep 2018, 02:10
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7
00:00

Difficulty:

85% (hard)

Question Stats:

39% (02:29) correct 61% (02:12) wrong based on 56 sessions

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The corners of an equilateral triangle are rounded to form semi-circles of radius 1. If the length of a side of the triangle before rounding is 10, what is the area of the resulting figure?

A. $$25\sqrt{3} + 3$$

B. $$25\sqrt{3} + \frac{3\pi}{2}$$

C. $$25\sqrt{3}$$

D. $$22\sqrt{3} + 3$$

E. $$22\sqrt{3} + \frac{3\pi}{2}$$

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The corners of an equilateral triangle are rounded to form semi-circle  [#permalink]

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Updated on: 28 Sep 2018, 05:57
Bunuel wrote:
The corners of an equilateral triangle are rounded to form semi-circles of radius 1. If the length of a side of the triangle before rounding is 10, what is the area of the resulting figure?

A. $$25\sqrt{3} + 3$$

B. $$25\sqrt{3} + \frac{3\pi}{2}$$

C. $$25\sqrt{3}$$

D. $$22\sqrt{3} + 3$$

E. $$22\sqrt{3} + \frac{3\pi}{2}$$

Area of the resulting figure = Area of Bigger equilateral - (Area cut out)

Area cut out from one corner = (1/3)*(Area of Equilateral triangle with side 2√3 - Area of Circle with radius 1) .

[The inscribed circle with radius 1 will give us the side of equilateral as 2√3 using 30-60-90 property)

Area cut out from one corner $$= (1/3)*[(√3/4)* 2√3)^2 - π1^2) = √3-(π/3)$$ .
Total Area cut out $$= 3*[√3-(π/3)] = 3√3 - π$$

Area of the resulting figure = $$(√3/4)* 10)^2 - (3√3 - π) = 22√3 +π$$

I think answer must be option E but highlighted part is still a bit confusing. Taking semicircle out from corners will involve different calculations.
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Originally posted by GMATinsight on 14 Sep 2018, 02:30.
Last edited by GMATinsight on 28 Sep 2018, 05:57, edited 1 time in total.
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Re: The corners of an equilateral triangle are rounded to form semi-circle  [#permalink]

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14 Sep 2018, 20:42
1
got the answer correct but took 3 mins first find the area of remaining figure = root3/4 10*10 - 3*root3/4*2*2( this is the area of equi triangle formed by making a semi circle and then add area of semi circle to remaining fig (pi*1^2/2 )*3
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Re: The corners of an equilateral triangle are rounded to form semi-circle  [#permalink]

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28 Sep 2018, 04:37
GMATNinja
Any better way?
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Joined: 15 Sep 2014
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The corners of an equilateral triangle are rounded to form semi-circle  [#permalink]

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30 Sep 2018, 21:08
Try to work out the sides of the big triangles and small triangle (pyto theorem) first.

(a) Area of the big equilateral triangle = (10/2) * 5 (3^(1/2)) = 25(3^(1/2))
(b) Area of 1 semi-circle = π (1^2) /2 = π/2
(c) Area of 1 small equilateral triangle= (3^(1/2))
(d) Area of 3 cut-out parts= (c)- (b) = 3*[(3) - (1) ]= 3[(3^(1/2)) - (π/2)] = 3(3^(1/2)) - 3π/2
Answer = (d) - (a) = 25(3^(1/2)) - [3(3^(1/2)) - 3π/2] = 22(3^(1/2))+ (3π/2)------ (E)
The corners of an equilateral triangle are rounded to form semi-circle &nbs [#permalink] 30 Sep 2018, 21:08
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