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# A polygon has 90 diagonals. How many sides does it have?

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Math Expert
Joined: 02 Aug 2009
Posts: 7334
A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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29 Apr 2016, 23:26
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Difficulty:

25% (medium)

Question Stats:

75% (01:32) correct 25% (01:40) wrong based on 163 sessions

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A polygon has 90 diagonals. How many sides does it have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
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Re: A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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30 Apr 2016, 00:41
1
1
The best way to solve this problem is to use the formula: An n sided polygon can have n*(n - 3)/2 diagonals

n*(n - 3)/2 = 90
n*(n - 3) = 180

Substitute n from the answer choices. n = 15

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Joined: 12 Jun 2015
Posts: 79
Re: A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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30 Apr 2016, 22:52
2
To take Vyshak point forward,

the formula for diagonals of an n sided polygon = n*(n - 3)/2 diagonal can be easily derived as follows:

consider a rectangle : 4 sides ABCD
now we know that there are just 2 diagonals here
but to apply the formula, we need to understand it
since there are 4 sides, n=4
but if you take a vertex, say A, it cannot have a diagonal with vertex B or vertex C (the adjacent vertices)
A can form a diagonal only with D
and B can form it with only B
hence when we calculate the no. of diagonals , we multiply sides n with (n-3) {These 3 include the adjacent vertices and point itself}
And we divide the product by 2 to avoid counting the diagonals twice

Here, in this particular problem, when you get the number of diagonals as 90
You can easily use this shortcut : n*(n - 3)/2 = 90
n*(n - 3) = 180 =15 * 12
n=15

Correct Option : D
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A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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26 Sep 2016, 06:43
No diagonals: nc2(total no of lines that can be drawn using n points)-n(no of sides)=n*(n-3)/2
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Re: A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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06 Mar 2018, 15:59
No of sides which can give the value as 180

there are two possibilities of getting 0 at the end..

13 and 15..hence substitute the same in the formula No of diagonals = n(n-3)/2..We will get answer with sub 15.
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Re: A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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08 Mar 2018, 07:23
The number of diagonals must be an integer. 15 is the only factor of 90 in the answer set.
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Re: A polygon has 90 diagonals. How many sides does it have?  [#permalink]

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12 Mar 2018, 08:47
chetan2u wrote:
A polygon has 90 diagonals. How many sides does it have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

If we let n = the number of sides, we can use the formula:

[n(n - 3)]/2 = number of diagonals

[n(n - 3)]/2 = 90

n^3 - 3n = 180

n^3 - 3n - 180 = 0

(n - 15)(n + 12) = 0

n = 15 or n = -12

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Re: A polygon has 90 diagonals. How many sides does it have?   [#permalink] 12 Mar 2018, 08:47
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