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Hexagon

by faraz_jeddah » Fri Nov 01, 2013 9:39 am
My approach

r = 2

Interior angle = ((6-2)/6) * 180 = 120

each arc represents 120/360 of the circle i.e. 1/3 rd

Length of one arc = 2*pi*2 * (1/3) = 4*pi / 3

Each vertice has 2 arcs.
=> 6 vertices have 12 arcs

Total length = 12 * 4*pi / 3 = 16*pi

OA is [spoiler]8*pi[/spoiler]
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by theCodeToGMAT » Fri Nov 01, 2013 9:50 am
I got 8*pi

Angle of Hexagon = (n-2)*180 = (6-2)*180 = 720

Measure of each angle = 720/6 = 120

Measure of one arc of 120 degree = 120/360 * 2 * pi * (2) = 4/3 * pi

Total sum of arcs = 6 [4/3 * pi] = 8 * pi
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by faraz_jeddah » Fri Nov 01, 2013 10:00 am
theCodeToGMAT wrote:I got 8*pi

Angle of Hexagon = (n-2)*180 = (6-2)*180 = 720

Measure of each angle = 720/6 = 120

Measure of one arc of 120 degree = 120/360 * 2 * pi * (2) = 4/3 * pi

Total sum of arcs = 6 [4/3 * pi] = 8 * pi
I see you edited your answer :)

Mind telling me how you got 6 arcs and not 12.
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by theCodeToGMAT » Fri Nov 01, 2013 10:06 am
faraz_jeddah wrote:
theCodeToGMAT wrote:I got 8*pi

Angle of Hexagon = (n-2)*180 = (6-2)*180 = 720

Measure of each angle = 720/6 = 120

Measure of one arc of 120 degree = 120/360 * 2 * pi * (2) = 4/3 * pi

Total sum of arcs = 6 [4/3 * pi] = 8 * pi
I see you edited your answer :)

Mind telling me how you got 6 arcs and not 12.
Yep, Earlier I made a silly mistake while solving .. I realized & corrected it

Check the attachment for the reason :) .. Each arc is made with different color..
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by GMATGuruNY » Fri Nov 01, 2013 10:56 am
theCode's figure is great, but on the GMAT we won't have access to different colors.
To avoid careless errors in geometry problems:

1. DRAW your own figure.
2. Label EVERYTHING.

Image

Hexagon ABCDEF:
In a polygon with n sides, the sum of the interior angles = (n-2)180.
In a hexagon, the sum of the interior angles = (6-2)180 = 720.
In a REGULAR hexagon, all of the sides are equal, as are all of the interior angles.
Thus, each of the interior angles of hexagon ABCDEF = 720/6 = 120.

Circle with center B:
Since r=2, the circumference = 2Ï€r = 2*Ï€*2 = 4Ï€.
Since ∠ABC = 120, and 120/360 = 1/3, the arc intercepted by ∠ABC -- arc AC -- constitutes 1/3 of the circumference.
Thus, arc AC = (1/3) * 4Ï€ = (4/3)Ï€.

Sum of the arc lengths:
List all of the arcs.
Each vertex is the endpoint of 2 different arcs.
Thus, each vertex must appear in our list EXACTLY 2 TIMES:
AC
AE
BD
BF
CE
DF
Total = 6.
Sum of the arc lengths = 6 * (4/3)Ï€ = 8Ï€.

The correct answer is C.
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by mevicks » Fri Nov 01, 2013 8:08 pm
Image
Each vertex of the regular hexagon shown above is the center of a circle. If the side of the hexagon is 2, what is the total length of the arcs?
A) 4Ï€
B) 6Ï€
C) 8Ï€
D) 12Ï€
E) 16Ï€
Regular hexagon can be broken up into 6 equilateral triangles.

This property is very helpful during time crunch situations and thus we can play around with regular hexagons very easily and reach a conclusion just by looking at the symmetry (the mathematical proof can be found out but is not necessary)

Focus on any one equilateral triangle and its corresponding arc:
Image

Length of the arc = (60/360)2π2 = π(2/3)
We have 12 such arcs encompassing the equilateral triangles, thus the total length = 12Ï€(2/3) = [spoiler]8Ï€[/spoiler]

Answer C
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by mevicks » Fri Nov 01, 2013 8:12 pm
mevicks wrote:
Regular hexagon can be broken up into 6 equilateral triangles.

This property is very helpful during time crunch situations and thus we can play around with regular hexagons very easily and reach a conclusion just by looking at the symmetry (the mathematical proof can be found out but is not necessary)
Also, such properties are useful on complex problems:
https://www.beatthegmat.com/hexagon-prob ... 06401.html
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