The number of diagonals D of a polygon with n sides is given...

The number of diagonals $D$ of a polygon with $n$ sides is given by the formula
$$D=\frac{n}{2}(n-3)$$
a. Determine the number of sides of a polygon with 464 diagonals.
b. Can a polygon have 12 diagonals? Explain.

Key Concepts

Quadratic Equations

When the polygon diagonals formula is set equal to a specific number of diagonals, solving for n typically results in a quadratic equation. Techniques such as factoring, completing the square, or applying the quadratic formula are used to determine the possible values of n. This algebraic process is essential for finding valid solutions within the context of geometric problems.

Integer Constraints in Polygon Geometry

In geometric contexts, especially with polygons, the number of sides (n) must be a whole number greater than or equal to 3. This integer constraint means that when solving the quadratic equation derived from the diagonals formula, only integer solutions that satisfy the condition n ? 3 are acceptable. Such constraints ensure that the theoretical solution corresponds to a realizable polygon.

Polygon Diagonals Formula

This formula, D = n(n - 3)/2, establishes a direct relationship between the number of sides (n) of a polygon and the number of its diagonals (D). It is derived from the observation that each vertex can connect to (n - 3) other vertices (excluding itself and its two adjacent vertices), and dividing by 2 corrects for the double counting of each diagonal. This formula is a fundamental tool in polygon geometry.

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