If a Polygon has 44 Diagonals, Then How Many Sides are There in the Polygon?

Question: If a polygon has 44 diagonals, then how many sides are there in the polygon?

1. 8
2. 9
3. 10
4. 11
5. 12

“If a Polygon has 44 Diagonals, Then How Many Sides are There in the Polygon?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "Kaplan's GMAT Math Workbook, 10th Edition". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

It is given in the question that a polygon has 44 diagonals.
It is asked in the question to find out the no of sides of the polygon.
|Number of diagonals in an n-sided polygon = nC2-n

=> nc2 - n = 44

=> n!/2!(n-2)! - n = 44

=> n(n-1)/2 - n = 44

Plug in the options
At n = 11 LHS = 11*10/2 - 11 = 55-11 = 44 = RHS
=>n = 11

Option D is the correct option.

Approach Solution 2:

It is given in the question that a polygon has 44 diagonals.
It is asked in the question to find out the no of sides of the polygon.
Number of diagonals in an n-sided polygon
= n(n-3)/2
Plug in the values of n to see where it equals 44

=> n = 11
Option D is the correct option.

Approach Solution 3:

If there are n sides, then there are n vertices. As a diagonal is formed using 2 vertices, each vertex of a polygon would make diagonals with each of the vertexes except the ones beside itself on either side.
So, for a diagonal to form a polygon must have more than 3 vertices. Therefore, there are n(n-3) diagonals that are formed using n vertices in n-sided polygon.
However, each diagonal is formed using 2 vertices and hence they are repeated twice.

Thus, total number of diagonals = n(n−3)/2 = 44
n = -3 and n = 11

Answer D.

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