If a regular polygon has 20 diagonals, then the sum of its interior angles is:

Question

Examoogle · Accepted Answer

The correct answer is Option A: $1080^\circ$. Use the diagonal formula to find $n$ by testing factors of 40 (the product of $n(n-3)$). For $n=8$, $8(5)/2 = 20$. Then, calculate $(8-2) 	imes 180^{\circ}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the diagonal formula to find $n$ by testing factors of 40 (the product of $n(n-3)$). For $n=8$, $8(5)/2 = 20$. Then, calculate $(8-2) 	imes 180^{\circ}$.

Formula Used: $$D = \frac{n(n-3)}{2}$$, $$	ext{Sum of interior angles} = (n-2) 	imes 180^{\circ}$$

Given: A regular polygon has 20 diagonals.

Steps: Calculate the number of sides (n): The number of diagonals $D$ in a polygon with $n$ sides is given by $D = \frac{n(n-3)}{2}$. Setting $D = 20$, we solve for $n$: $20 = \frac{n(n-3)}{2} \Rightarrow 40 = n^2 - 3n \Rightarrow n^2 - 3n - 40 = 0$.. Calculate sum of interior angles: The formula for the sum of interior angles of a polygon with $n$ sides is $(n-2) 	imes 180^{\circ}$. Since $n=8$:

Answer: A is correct because for a polygon with 20 diagonals, the number of sides $n$ is 8, and the sum of interior angles is $(8-2) 	imes 180^{\circ} = 1080^{\circ}$.

Exam Tip: This connects to coordinate geometry where internal angles of polygons are used to define symmetry and orientation.

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