Hexagon

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

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.

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..

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.



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 = 2πr = 2*π*2 = 4π.
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) * 4π = (4/3)π.

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)π = 8π.

The correct answer is C.

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:


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

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