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# What is the area of the shaded portion in circle O, as pictured above,

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Manager
Joined: 06 Jun 2014
Posts: 88
Location: United States
Concentration: Finance, General Management
GMAT 1: 450 Q27 V21
GPA: 3.47
What is the area of the shaded portion in circle O, as pictured above,  [#permalink]

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06 Feb 2016, 16:17
1
1
00:00

Difficulty:

25% (medium)

Question Stats:

84% (01:26) correct 16% (03:07) wrong based on 64 sessions

### HideShow timer Statistics

What is the area of the shaded portion in circle O, as pictured above, with m< O=60° and radius r = 10?

A) $$\frac{50}{3}$$ $$\pi$$ − $$50\sqrt{3}$$

B) $$\frac{50}{3}$$ $$\pi$$ − 50

C) $$\pi \sqrt{3}$$

D) $$\frac{50}{3}$$ $$\pi$$ − $$25\sqrt{3}$$

E) $$\frac{50}{3}$$ $$\pi$$

Attachments

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Intern
Joined: 03 Feb 2016
Posts: 10
Re: What is the area of the shaded portion in circle O, as pictured above,  [#permalink]

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06 Feb 2016, 17:31
It is not a hard problem, if you know the Area of a circle segment and the value of sin for a 60 degree angle.

A=$$\frac{r^2}{2}$$*($$\frac{\pi*\alpha}{180}$$-sin$$\alpha$$)

Sin $$\alpha$$ 60 = $$\frac{\sqrt{3}}{2}$$

Another way is to find the Area of the triangle and the Area of the sector of circle to find the Area of the Segment of the Circle.

Area Sector - Area Triangle = Area Segment.

Hope it helps.

Ans. D)
Intern
Joined: 14 Jul 2015
Posts: 22
GMAT 1: 680 Q44 V40
GMAT 2: 710 Q49 V37
What is the area of the shaded portion in circle O, as pictured above,  [#permalink]

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07 Feb 2016, 01:32
1
As mentioned by FelixM, you simply need to take the area of the sector/circle slice and subtract the area of the interior triangle.

Area of the circle slice:

1. $$\frac{60°}{360°} * π 10 ^2$$
2. (1) reduces into $$\frac{{π 50}}{3}$$

Area of the triangle:
1. We see that the interior angle of our triangle is 60°. We are dealing with an equilateral triangle.
2. We bisect our triangle down the middle to find our height. When we bisect the triangle, that creates two sub-triangles, mirrored, each conforming to the 30-60-90 special triangle. Plugging in the magic numbers for this triangle gives us a triangle height of $$5\sqrt{3}$$.
3. $$5 \sqrt{3} * 10 * \frac{1}{2} = 25 \sqrt{3}$$

To find the area of the shaded portion:
1. As mentioned above, subtract the area of the triangle from the area of the sector: $$\frac{{π 50}}{3} - 25\sqrt{3}$$, which is option D.
Manager
Joined: 06 Jun 2014
Posts: 88
Location: United States
Concentration: Finance, General Management
GMAT 1: 450 Q27 V21
GPA: 3.47
Re: What is the area of the shaded portion in circle O, as pictured above,  [#permalink]

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09 Feb 2016, 17:47
1
There are only two formulas we need to know for this question:
1) Area of Sector: $$\frac{n}{360}$$ * $$\pi$$ $$r^2$$
2) Area of an Equilateral Triangle: $$\frac{Side^2 \sqrt{3}}{4}$$

Area of Sector - Area of an Equilateral

$$\frac{60 \pi 10^2}{360} - \frac{10^2 \sqrt{3}}{4}$$

$$\frac{50 \pi}{3} - 25\sqrt{3}$$
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Joined: 09 Sep 2013
Posts: 9836
Re: What is the area of the shaded portion in circle O, as pictured above,  [#permalink]

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26 Jan 2019, 16:59
Hello from the GMAT Club BumpBot!

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Re: What is the area of the shaded portion in circle O, as pictured above,   [#permalink] 26 Jan 2019, 16:59
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