the sum of the interior angles of a regular polygon is twice the sum of the exterior angles. find the number of sides of the polygon

The sum of the interior angles of a polygon is given by the formula:

\[
S_{\text{interior}} = 180(n-2)
\]

where \(n\) is the number of sides of the polygon.

The sum of the exterior angles of any polygon is always \(360^\circ\), regardless of the number of sides. So, the sum of the exterior angles is:

\[
S_{\text{exterior}} = 360^\circ
\]

According to the problem, the sum of the interior angles is twice the sum of the exterior angles. Therefore, we can set up the following equation:

\[
180(n-2) = 2 \times 360
\]

Simplifying the equation:

\[
180(n-2) = 720
\]

Divide both sides by 180:

\[
n - 2 = 4
\]

Add 2 to both sides:

\[
n = 6
\]

Thus, the polygon has **6 sides**. It is a regular hexagon.

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Would you like more details or have any questions on this solution?

Here are some related questions to explore:

1. How do you calculate the sum of the interior angles for any polygon?
2. What is the sum of the interior angles for a decagon (10-sided polygon)?
3. How do the interior and exterior angles of a polygon relate to each other in general?
4. What is the measure of each interior angle in a regular hexagon?
5. What are the applications of polygons with different numbers of sides in geometry?

**Tip:** For any regular polygon, the exterior angle is simply \( \frac{360^\circ}{n} \), where \(n\) is the number of sides.

The sum of the interior angles of a regular polygon is twice the sum of the exterior angles. Find the number of sides of the polygon.

Discover how to determine the number of sides of a regular polygon when the sum of the interior angles is twice the sum of the exterior angles. This problem introduces polygon geometry, interior and exterior angle sums, and applies key geometric formulas. Ideal for students in Grades 8-10.

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