Final answer:
A regular polygon with an interior angle of 178° has 180 sides, calculated by using the sum of exterior angles which always equals 360°.
Step-by-step explanation:
The question is asking to calculate the number of sides of a regular polygon with an interior angle of 178°. To find the number of sides of a regular polygon, we use the formula for the sum of interior angles, which is (n - 2) × 180°, where n is the number of sides in the polygon. Since we know the measure of one interior angle, we can rearrange this formula to solve for n.
The sum of the interior angles of a polygon can also be calculated by multiplying the number of triangles that can be drawn within the polygon (which is n - 2) by 180°. Each triangle has a sum of 180°, as mentioned in LibreTexts™.
To find the number of sides, we do the following:
- Calculate the measure of the exterior angle of the polygon by subtracting the interior angle from 180°. So, the exterior angle = 180° - 178° = 2°.
- Use the fact that the sum of all exterior angles of a polygon is always 360°. Thus, divide 360° by the measure of one exterior angle to find the number of sides: 360° / 2° = 180.
Therefore, a regular polygon with an interior angle of 178° has 180 sides.