A polygon has 12 edges. How many different diagonals does it have?

Question

(A) 54
(B) 66
(C) 108
(D) 132
(E) 144

generis · Accepted Answer

The number of diagonals for a polygon, where n is the number of sides, is found by the formula

\frac{(n)(n - 3)}{2}

\frac{(12)(9)}{2}  = 54

Answer A

BrentGMATPrepNow · Answer

The polygon in question has 12 vertices.

Let's focus on ONE VERTEX, which we'll call vertex A
How many diagonal can be drawn from vertex A? 
We'll there are 11 other vertices to connect to, HOWEVER the vertices on either side of vertex A are out of contention, since joining two adjacent points will not create a diagonal. 
So, we can create a diagonal with vertex A by connecting to any of the  9  eligible vertices.

Using the same logic, we can conclude that we can create a diagonal with vertex B by connecting to any of the  9  eligible vertices.

And we can create a diagonal with vertex C by connecting to any of the  9  eligible vertices.

There are 12 vertices, so the TOTAL number of diagonals = (12)( 9 ) = 108
HOWEVER, this is not the final answer, since we have counted each diagonal TWICE. 
For example, in the first step, we counted the diagonal created by joining vertex A with vertex B, and in the second step, we counted the diagonal created by joining vertex B with vertex A

To account for this duplication, we must take 108 and divide it by 2 to get: 54

Answer:  A

Cheers,
Brent

Kinshook · Answer

Given: A polygon has 12 edges.

Asked: How many different diagonals does it have?

Number of edges = Number of sides = n = 12

The number of diagonals = n(n-3)/2 = 12*9/2 = 54

IMO A

Camach700 · Answer

A general and logical formula can be derived by using combinations:

Combinations formula : n! /

#total diagonals (couples of vertices) = n! /   ; where n is the number of sides and k equals 2 (grouping each possible couple)

Then we could subtract the number of sides of the polygon, as those couples would not represent diagonals.

In this case we would have

#total diagonals (sides included)  = 12! /   = 66

#total diagonals (sides excluded)  = 66 - 12 = 54

nbrn226 · Answer

12 edges means 12 vertices.
Number of lines that can be drawn joining 2 vertices =12C2
Out of which 12 are edges, rest are diagonals.
So the no of diagonals =12C2-12=12*11÷2 - 12=66-12=54

Posted from my mobile device

Aboyhasnoname · Answer

For those who are weak in quant like me, and can't possibly know the diagonals in a polygon formula....

Approach 1. 
There are 12 vertices....See how many lines are connecting....Pick any 2..12C2..You will get a. total of 66 lines connecting all the dots with each other.. 
Now in 62 there are 12 lines that are sides ..so remove these 12...66-12...54.....

Approach 2. 
Each point can connect with 9 other points to form a diagonal(Can't connect with itself, and the two on sides)..So a total of 9 options it has to connect..
12 Points will connect to 12 X 0 = 108...
But in this..You have counted each diagonal twice...Let vertices be A and E...So you have counted AE and EA as separate where as they are the same... 
So divide this 108 by 2. you get 108...

Approach 3. The formula approach {n(n-3)}/2 ...12 X 9/2.....This is basically same as 2..but in the formula form.....

A polygon has 12 edges. How many different diagonals does it have?

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A polygon has 12 edges. How many different diagonals does it have?

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