Each interior angle of a regular polygon is thrice of the exterior polygon. find the number of sides of the polygon

General
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each interior angle of a regular polygon is thrice of the exterior polygon. find the number of sides of the polygon.

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LectureNotes said each interior angle of a regular polygon is thrice of the exterior polygon. Find the number of sides of the polygon.

Answer: To determine the number of sides of a regular polygon where each interior angle is thrice the measure of each exterior angle, we can follow these steps:

  1. Understand the relationship between interior and exterior angles:

    • For any polygon, the sum of an interior angle and its corresponding exterior angle is always 180^\circ.
    • Let the measure of each exterior angle be E.
    • Then, the measure of each interior angle will be 3E (since it is given to be thrice the exterior angle).

  2. Set up the equation:

    • Since the interior and exterior angles are supplementary, we have:
      E + 3E = 180^\circ - Simplify the equation:
      4E = 180^\circ
    • Solving for E gives:
      E = \frac{180^\circ}{4} = 45^\circ

  3. Find the number of sides of the polygon:

    • The exterior angle of a regular polygon is given by \frac{360^\circ}{n}, where n is the number of sides.
    • We know that each exterior angle E = 45^\circ.
    • Therefore, we can set up the equation:
      \frac{360^\circ}{n} = 45^\circ
    • Solving for n:
      n = \frac{360^\circ}{45^\circ} = 8

Thus, the number of sides of the polygon is \boxed{8}.