Math Problem Statement

Find the number of sides in an irregular polygon with one exterior angle of 120° and all others 12°.

Solution

We are asked to find the number of sides in an irregular polygon where one exterior angle is 120120^\circ, and all other exterior angles are 1212^\circ.

Key Concepts:

  1. Sum of exterior angles in a polygon: The sum of the exterior angles of any polygon is always 360360^\circ, regardless of whether it is regular or irregular.

  2. Approach:

    • Let nn be the total number of sides of the polygon.
    • We know that one exterior angle is 120120^\circ, and the remaining (n1)(n-1) exterior angles are each 1212^\circ.

Step-by-step Solution:

  • The sum of all exterior angles is given by: 120+(n1)×12=360120^\circ + (n-1) \times 12^\circ = 360^\circ

  • Simplifying the equation: 120+12(n1)=360120^\circ + 12^\circ(n - 1) = 360^\circ 120+12n12=360120^\circ + 12^\circ n - 12^\circ = 360^\circ 12n+108=36012^\circ n + 108^\circ = 360^\circ 12n=36010812^\circ n = 360^\circ - 108^\circ 12n=25212^\circ n = 252^\circ n=25212=21n = \frac{252^\circ}{12^\circ} = 21

Final Answer:

The polygon has 21 sides.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

  1. What is the relationship between exterior angles and the number of sides in a polygon?
  2. How would the formula change if all the angles were equal?
  3. Can you explain how the sum of exterior angles is always 360360^\circ?
  4. What is the formula for finding the number of sides in a regular polygon if all exterior angles are given?
  5. How does the irregularity of the polygon affect the calculation?

Tip: For any polygon, the sum of exterior angles is always 360360^\circ, whether the polygon is regular or irregular!

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygons
Exterior Angles

Formulas

Sum of exterior angles of a polygon = 360°

Theorems

The sum of exterior angles of any polygon is always 360°

Suitable Grade Level

Grades 9-12