Find the number of diagonals that can be drawn in an n-side polygon

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SUMMARY

The number of diagonals in an n-sided polygon is calculated using the formula n(n-3)/2. A recursive approach to derive this formula is established with D_{n+1} = D_n + n - 2, where D_n represents the number of diagonals in an n-gon. Understanding the reasoning behind this recursion is crucial, particularly in recognizing that counting diagonals from a single vertex leads to overcounting. This discussion clarifies the correct method to derive the total number of diagonals in polygons.

PREREQUISITES
  • Understanding of polygon properties and definitions
  • Familiarity with basic algebra and recursion
  • Knowledge of combinatorial mathematics
  • Ability to manipulate mathematical formulas
NEXT STEPS
  • Study the derivation of the diagonal formula for polygons
  • Explore recursive functions in mathematical contexts
  • Learn about combinatorial counting techniques
  • Investigate geometric properties of polygons
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Mathematicians, educators, students studying geometry, and anyone interested in combinatorial mathematics will benefit from this discussion.

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Find the number of diagonals that can be drawn in an n-side polygon.

The answer is n(n-3)/2.

I don't know how to do that.
 
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Set up a recursion for the number of diagonals in an n-gon: D_{n+1} = D_n + n - 2.
 
I don't understand how can I set up
D_{n+1} = D_n + n - 2.
 
How many diagonals can be drawn from 1 vertex? If you multiply that by the number of vertices you will get the wrong answer! Do you see why? How can you fix it?
 

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