Each interior angle of a regular polygon with 10 sides measures 144 degrees.
Understanding Regular Polygons and Their Angles
A regular polygon is a two-dimensional shape where all sides are of equal length and all interior angles are of equal measure. The measure of these angles is crucial for various geometric and architectural applications. For a regular polygon with 'n' sides, there are precise formulas to determine the measure of each interior angle.
Calculating the Interior Angle of a Decagon
A polygon with 10 sides is known as a decagon. To find the measure of each interior angle, we can use a standard formula.
The Formula for Interior Angle
The measure of each interior angle of a regular polygon can be calculated using the formula:
Each Interior Angle = (2n - 4) / n right angles
Where 'n' is the number of sides of the polygon.
Alternatively, since 1 right angle = 90 degrees, the formula can be expressed in degrees as:
Each Interior Angle = ((n - 2) × 180°) / n
Let's apply the first formula provided for a 10-sided polygon (n=10):
-
Substitute n = 10 into the formula:
Each interior angle = (2 × 10 - 4) / 10 right angles -
Simplify the numerator:
Each interior angle = (20 - 4) / 10 right angles
Each interior angle = 16 / 10 right angles -
Convert right angles to degrees:
Since 1 right angle = 90°,
Each interior angle = (16 / 10) × 90°
Each interior angle = 1.6 × 90°
Each interior angle = 144°
Step-by-Step Calculation Table
| Step | Calculation | Result |
|---|---|---|
| 1. Identify Number of Sides (n) | n = 10 | 10 |
| 2. Apply Formula (in right angles) | (2n - 4) / n | (2*10 - 4) / 10 |
| 3. Simplify Numerator | (20 - 4) / 10 | 16 / 10 |
| 4. Express in Decimal (Right Angles) | 1.6 right angles | 1.6 right angles |
| 5. Convert to Degrees | 1.6 × 90° | 144° |
Related Concepts and Properties
Understanding the interior angles of regular polygons also involves knowledge of other geometric properties:
- Sum of Interior Angles: The sum of all interior angles of a polygon can be found using the formula:
(n - 2) × 180°. For a decagon, this would be(10 - 2) × 180° = 8 × 180° = 1440°. Dividing this sum by the number of sides (10) gives the individual angle:1440° / 10 = 144°. - Exterior Angles: An exterior angle of a regular polygon is
360° / n. For a decagon, each exterior angle is360° / 10 = 36°. - Relationship between Interior and Exterior Angles: The interior angle and its adjacent exterior angle always sum to 180°. In this case, 144° + 36° = 180°, which confirms our calculations.
This consistent relationship provides a valuable way to verify calculations for polygon angles.
