Determining the Number of Sides in a Polygon Given the Sum of Interior Angles
Geometry problems often involve understanding the properties of polygons, particularly the sum of their interior angles. If you're given that the sum of the interior angles in a regular polygon is 2340°, how can you determine the number of sides? This article will walk you through the process step by step, using a variety of formulas and equations.
Understanding the Sum of Interior Angles
The sum of the interior angles of a polygon with n sides is given by the formula:
S 180(n - 2)
This formula is derived from the fact that any polygon can be divided into n - 2 triangles, each contributing 180° to the total sum of the angles.
Applying the Formula
Given that the sum of the interior angles is 2340°, we set up the equation as follows:
S 180(n - 2) 2340°
First, we solve for n:
2340 180(n - 2)
Divide both sides by 180:
n - 2 frac{2340}{180} 13
Add 2 to both sides to find n:
n 13 2 15
Therefore, the polygon has 15 sides.
Verifying with Exterior Angles
Another way to verify is by using the exterior angles of a regular polygon. Each exterior angle in a regular polygon is given by:
frac{360°}{n}
Since the sum of the interior and exterior angles at each vertex is 180°, each interior angle is:
180° - frac{360°}{n}
The total sum of the interior angles is then:
n(180° - frac{360°}{n}) 180n - 360°
Setting this equal to 2340°, we solve for n again:
180n - 360 2340
Add 360 to both sides:
180n 2700
Divide by 180:
n frac{2700}{180} 15
This confirms our previous result.
Conclusion
Understanding the properties of polygons, specifically the sum of their interior angles, can help solve a variety of geometry problems. The process involves setting up and solving an equation, which can be done in multiple ways. Whether you use the formula for the sum of interior angles or consider the relationship with exterior angles, the method is clear and straightforward. Knowing these formulas and how to apply them is key to mastering geometry problems.