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A certain game board is in the shape of a non-convex [#permalink]

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20 Dec 2006, 14:41

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A certain game board is in the shape of a non-convex polygon, with spokes that extend from each vertex to the center of the board. If each spoke is 8 inches long, and spokes are used nowhere else on the board, what is the sum of the interior angles of the polygon?

(1) The sum of the exterior angles of the polygon is 360º.

(2) The sum of the exterior angles is equal to five times the total length of all of the spokes used.

The formula for the sum of the interior angles of a non-convex polygon is (n â€“ 2)(180), where n represents the number of sides. To find the sum of the interior angles of the polygon then, we need to know the number of sides. We can therefore rephrase the question:

How many sides does the game board have?

(1) INSUFFICIENT: It tells us nothing about the number of sides. The sum of the exterior angles for any non-convex polygon is 360.

(2) SUFFICIENT: The sum of the exterior angles = 5 Ã— length of each spoke Ã— number of spokes.

360 = 5(8)(x)
360 = 40x
9 = x

The game board has nine sides. The sum of its interior angles is (9 â€“ 2)(180) = 1260.

Re: A certain game board is in the shape of a non-convex [#permalink]

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30 Sep 2015, 21:46

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Good Question,

In case of a concave (non-convex) polygon, you subtract exterior angle from the equation (of sum).

E.g. check out attached image as an illustration.

JusTLucK04 wrote:

How can the sum of all exterior angles of a non convex polygon be 360..I take a polygon with n sides and the sides are so acute tat exterior angle is close to 360..Imagine something like the shape of a star just with more side..

Re: A certain game board is in the shape of a non-convex [#permalink]

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09 Aug 2015, 22:23

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A certain game board is in the shape of a non-convex [#permalink]

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30 Sep 2015, 21:41

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Sum of a convex polygon is always 360.

An exterior angle of a polygon is an angle that forms a linear pair with one of the angles of the polygon.

Two exterior angles can be formed at each vertex of a polygon. The exterior angle is formed by one side of the polygon and the extension of the adjacent side. For the hexagon shown at the left, <1 and <2 are exterior angles for that vertex. Be careful, as <3 is NOT an exterior angle.

How can the sum of all exterior angles of a non convex polygon be 360..I take a polygon with n sides and the sides are so acute tat exterior angle is close to 360..Imagine something like the shape of a star just with more side..

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Last edited by rohitmanglik on 30 Sep 2015, 21:53, edited 2 times in total.

A certain game board is in the shape of a non-convex [#permalink]

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05 Nov 2015, 07:47

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This is a lot easier than it sounds.

What is the sum of the interior angles of the polygon? ==> What is (n-2)*180? ==> What is n?

Statement 1: Supplies a known geometric fact that adds no new information. The sum of exterior angles of a polygon is always 360. This gives no information about n, insuff.

Statement 2: 5*n*8 = sum of ext angles ==> 40*n = 360 ==> n = 9. Suff.

from 2
The sum of the exterior angles is equal to five times the total length of all of the spokes used.
Let say Y= # of sides and Y sum of interior angles
for any type of polygon the interior angle and exterior angle always add to 180Â°.

5*8*N + Y = 180*N
=>40N+Y = 180 N
and
Y = (N-2)*180

from 2 The sum of the exterior angles is equal to five times the total length of all of the spokes used. Let say Y= # of sides and Y sum of interior angles for any type of polygon the interior angle and exterior angle always add to 180Â°.

5*8*N + Y = 180*N =>40N+Y = 180 N and Y = (N-2)*180

N = 9 and Y =(9-2)*180= 7*180 So B whats the OA?

But How did you get N=9 ?

40N+Y = 180 N ------ sum of exterior angles + sum of interior angles
Y = (N-2)*180 --------sum of interior angles

The formula for the sum of the interior angles of a non-convex polygon is (n â€“ 2)(180), where n represents the number of sides. To find the sum of the interior angles of the polygon then, we need to know the number of sides. We can therefore rephrase the question:

How many sides does the game board have?

(1) INSUFFICIENT: It tells us nothing about the number of sides. The sum of the exterior angles for any non-convex polygon is 360.

(2) SUFFICIENT: The sum of the exterior angles = 5 Ã— length of each spoke Ã— number of spokes.

360 = 5(8)(x) 360 = 40x 9 = x

The game board has nine sides. The sum of its interior angles is (9 â€“ 2)(180) = 1260.

The correct answer is B.

Got B too but in different way ;

(2) SUFFICIENT: The sum of the exterior angles = 5 Ã— length of each spoke Ã— number of spokes.

360 = 5(8)(x)
how did u get this 360 as sum of the exterior angles; (2) does not say that
unless you deduce it .

In my opinion the answer should be C. We don't know what is the sum of exterior angles. We have to use this information from the first statement,then only we can set up this equation
360=5(8x)
that is sufficient to know the number of sides and hence the sum of interior angles.

Convex polygon is a polygon inwhich each interior angle has a measure of less than 180.

Re: A certain game board is in the shape of a non-convex [#permalink]

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27 Apr 2014, 10:42

How can the sum of all exterior angles of a non convex polygon be 360..I take a polygon with n sides and the sides are so acute tat exterior angle is close to 360..Imagine something like the shape of a star just with more side..
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Re: A certain game board is in the shape of a non-convex [#permalink]

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08 Sep 2017, 10:12

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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