The value of 
x so that every regular polygon with more sides has non-integer interior angles

To find the value of \( x \), where \( x \) is the number of sides of a regular polygon such that every polygon with more sides has non-integer interior angles, we need to look at the formula for the interior angle of a regular polygon.

The interior angle \( A \) of a regular polygon with \( n \) sides is given by:

\[
A = \frac{(n-2) \cdot 180^\circ}{n}
\]

For every regular polygon with more than \( x \) sides to have a non-integer interior angle, we want to find the largest integer \( x \) such that the interior angle is still an integer.

1. Set the formula for the interior angle \( A \) to be an integer. This implies that the expression

\[
\frac{(n-2) \cdot 180}{n}
\]

must be an integer for \( n = x \).

2. Now, let's check small values of \( n \) to find the point at which this no longer results in an integer:

   - For \( n = 3 \) (triangle): 
     \[
     A = \frac{(3-2) \cdot 180}{3} = 60^\circ \quad \text{(integer)}
     \]
   - For \( n = 4 \) (square): 
     \[
     A = \frac{(4-2) \cdot 180}{4} = 90^\circ \quad \text{(integer)}
     \]
   - For \( n = 5 \) (pentagon): 
     \[
     A = \frac{(5-2) \cdot 180}{5} = 108^\circ \quad \text{(integer)}
     \]
   - For \( n = 6 \) (hexagon): 
     \[
     A = \frac{(6-2) \cdot 180}{6} = 120^\circ \quad \text{(integer)}
     \]
   - For \( n = 7 \) (heptagon): 
     \[
     A = \frac{(7-2) \cdot 180}{7} \approx 128.57^\circ \quad \text{(non-integer)}
     \]

Thus, the first polygon with a non-integer interior angle is the regular heptagon with \( n = 7 \) sides. So, the value of \( x \) is **6**. Any regular polygon with more than 6 sides will have a non-integer interior angle.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do exterior angles of polygons relate to their interior angles?
2. What is the sum of interior angles for any \( n \)-sided polygon?
3. Can the interior angle of a polygon ever be 180 degrees? Why or why not?
4. What is the formula for the area of a regular polygon in terms of its side length and number of sides?
5. How do the number of sides of a polygon affect the ratio of its perimeter to its area?

**Tip:** The sum of the exterior angles of any regular polygon is always \( 360^\circ \), no matter the number of sides.

The value of x so that every regular polygon with more sides has non-integer interior angles.

Discover how to determine the value of x, the number of sides of a regular polygon, such that every polygon with more sides has non-integer interior angles. This problem explores geometric principles and the formula for the interior angles of polygons, suitable for students in Grades 8-10.

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