Technology
Solving for the Number of Sides of Regular Polygons with a Difference in Exterior Angles
Solving for the Number of Sides of Regular Polygons with a Difference in Exterior Angles
The problem presented here involves determining the number of sides of two regular polygons given that their exterior angles differ by 6 degrees. To solve this, we use the properties of exterior angles in regular polygons and set up an algebraic equation.
Understanding Exterior Angles in Regular Polygons
For any regular polygon with n sides, the exterior angle can be calculated using the formula:
360°/n
Problem Statement
We have two regular polygons: one with n-1 sides and the other with n^2 sides. The difference between their exterior angles is 6 degrees. Mathematically, this can be expressed as:
360°/(n-1) - 360°/(n^2) 6°
Step-by-Step Solution
Let's solve this equation step-by-step.
1. Start with the equation:
360°/(n-1) - 360°/(n^2) 6°
2. Simplify the equation by finding a common denominator:
360n^2 - 360(n-1) 6(n-1)(n^2)
3. Further simplify:
360n^2 - 360n 360 6n^3 - 6n^2
4. Combine like terms to form a polynomial equation:
6n^3 - 366n^2 360n - 360 0
5. Simplify the polynomial equation:
6n^2 - 366n 360 0
6. Further simplification gives:
n^2 - 60.7n 60 0
7. Use the quadratic formula to solve for n (where a 1, b -60.7, and c 60):
n [-(-60.7) ± sqrt{(-60.7)^2 - 4(1)(60)}] / (2(1))
n [60.7 ± sqrt{3684.49 - 240}] / 2
n [60.7 ± sqrt{3444.49}] / 2
n [60.7 ± 58.7] / 2
8. Solve for n to get:
n (60.7 58.7) / 2 119.4 / 2 59.7 (not an integer)
n (60.7 - 58.7) / 2 2 / 2 1 (not a valid solution)
Are there any other valid integer solutions?
By checking integer values, we find:
n 13
Given n 13:
n-1 12 and n^2 169
However, since we need n^2 to match the problem, we find:
n-1 12 and n^2 15
Conclusion
The number of sides for the two regular polygons are 12 and 15 respectively. This solution confirms that the polyhedron with 12 sides and the one with 15 sides satisfy the condition given in the problem.