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The sum of the measures of the interior angles of a polygon with [tex]n[/tex] sides is

A. [tex](n-3) \times 180^{\circ}[/tex]
B. [tex](n-2) \times 180^{\circ}[/tex]
C. [tex](n-2) \times 360^{\circ}[/tex]
D. [tex](n-3) \times 3600^{\circ}[/tex]
E. [tex](n-2) \times 90^{\circ}[/tex]


Sagot :

To determine the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we can use a well-known formula in geometry. Here’s the step-by-step explanation:

1. Understanding the Problem:
- We need to find a formula that gives us the total sum of the interior angles of a polygon based on the number of its sides.

2. Concept of Polygon Interior Angles:
- A polygon with [tex]\( n \)[/tex] sides (an [tex]\( n \)[/tex]-gon) can be divided into [tex]\( n - 2 \)[/tex] triangles.
- This is because any polygon can be triangulated into [tex]\( n - 2 \)[/tex] non-overlapping triangles.

3. Sum of Angles in Triangles:
- We know the sum of the interior angles of a triangle is [tex]\( 180^\circ \)[/tex].

4. Relating Triangles to the Polygon:
- Since the polygon can be divided into [tex]\( n - 2 \)[/tex] triangles, the sum of the interior angles of the polygon is the sum of the interior angles of these [tex]\( n - 2 \)[/tex] triangles.

5. Calculation:
- Each triangle contributes [tex]\( 180^\circ \)[/tex] to the sum.
- Therefore, the sum is [tex]\( (n - 2) \times 180^\circ \)[/tex].

6. Conclusion:
- The sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula [tex]\( (n-2) \times 180^\circ \)[/tex].

Hence, the correct answer is:

[tex]\[ \boxed{(n-2) 180^\circ} \][/tex]