A polygon with interior angles that add up to 4140 degrees has 25 sides.
Understanding the Polygon Angle Sum Formula
The sum of the interior angles of any polygon, regardless of whether it is regular or irregular, can be determined using a fundamental formula. This formula establishes a direct relationship between the number of sides a polygon has and the total measure of its internal angles.
The formula is:
Sum = (n - 2) × 180°
Where:
Sumis the total measure of all interior angles.nis the number of sides of the polygon.
This formula is derived from the fact that any polygon can be divided into a specific number of non-overlapping triangles by drawing diagonals from one vertex. For an n-sided polygon, (n - 2) triangles can be formed. Since the sum of angles in a single triangle is always 180°, multiplying (n - 2) by 180° gives the total sum of all interior angles.
Calculating the Number of Sides
To find the number of sides (n) when the sum of interior angles is known, we can rearrange the formula. Given that the sum of the interior angles is 4140 degrees:
-
Set up the equation:
4140 = (n - 2) × 180 -
Divide both sides by 180:
4140 / 180 = n - 2
23 = n - 2 -
Add 2 to both sides to isolate 'n':
n = 23 + 2
n = 25
Therefore, a polygon whose interior angle measures add up to 4140 degrees has 25 sides. This calculation confirms that a polygon with 25 sides indeed has an interior angle sum of 4140 degrees.
Practical Application: Polygon Types and Angle Sums
Understanding the angle sum formula is crucial for classifying and analyzing polygons. The total interior angle measure uniquely determines the number of sides a polygon possesses.
Here's a quick reference for common polygons and their angle sums:
| Number of Sides (n) | Polygon Name | Sum of Interior Angles (n-2) × 180° |
|---|---|---|
| 3 | Triangle | (3-2) × 180° = 180° |
| 4 | Quadrilateral | (4-2) × 180° = 360° |
| 5 | Pentagon | (5-2) × 180° = 540° |
| 6 | Hexagon | (6-2) × 180° = 720° |
| ... | ... | ... |
| 25 | Icosikai-pentagon | (25-2) × 180° = 4140° |
While the sum of interior angles is fixed for a polygon with a given number of sides, the measure of individual angles can vary significantly in irregular polygons. For regular polygons, where all sides are equal in length and all interior angles are equal in measure, each interior angle can be found by dividing the total sum by the number of sides. For a regular 25-sided polygon (an icosikai-pentagon), each interior angle would measure 4140° / 25 = 165.6°.
