Understanding the Sides of a Regular Polygon with a Specific Interior Angle

Regular polygons are fascinating geometric shapes where all angles and sides are equal. Determining the number of sides using the interior angle is a common problem in geometry that can be solved using a simple formula. In this article, we will explore how to find the number of sides of a regular polygon when each interior angle is given, specifically when each interior angle is 135°. Along the way, we will also look at other methods and variations to solve similar problems.

Methods to Find the Number of Sides of a Regular Polygon

The formula for the interior angle of a regular polygon is given by:

Interior Angle (frac{(n-2) times 180^circ}{n})

Where:

(n): the number of sides of the polygon (180^circ): the sum of the angles in a triangle

Let's solve for (n) when the interior angle is (135^circ).

Step-by-Step Solution

Starting with the formula:

135° (frac{(n-2) times 180^circ}{n})

Multiplying both sides by (n):

135n (n-2) times 180

Expanding and rearranging the equation:

135n 180n - 360

360 180n - 135n

360 45n

Dividing both sides by 45:

(n frac{360}{45} 8)

This means a regular polygon with each interior angle of 135° has 8 sides. This polygon is known as an octagon.

Alternative Methods

Here are other methods to solve the same problem:

Method 1: Direct Algebraic Approach

Each interior angle of an (n)-sided polygon can also be expressed using the exterior angle. The exterior angle is the supplement of the interior angle, so:

Exterior Angle (180^circ - 135^circ 45^circ)

The total sum of the exterior angles of a polygon is 360°. Therefore:

Number of sides (frac{360^circ}{text{Exterior Angle}} frac{360^circ}{45^circ} 8)

Method 2: Using Exteriors Directly

If the interior angle is 150°, the exterior angle would be:

Exterior Angle (180^circ - 150^circ 30^circ)

Number of sides (frac{360^circ}{30^circ} 12)

Generalizing the Solution

The formula to find the number of sides (n) in a regular polygon given an interior angle (theta) is:

(n frac{360^circ}{180^circ - theta})

For example:

If (theta 120^circ): (n frac{360^circ}{180^circ - 120^circ} frac{360^circ}{60^circ} 6), resulting in a hexagon. If (theta 144^circ): (n frac{360^circ}{180^circ - 144^circ} frac{360^circ}{36^circ} 10), resulting in a decagon.

Conclusion and Applications

Understanding how to find the number of sides of a regular polygon using its interior angle is crucial in various fields, including architecture, engineering, and mathematics. Whether you are working with complex geometric shapes or simplifying real-world problems, knowing this formula can be invaluable.

By practicing with different interior angles, you can gain a deeper understanding of the relationship between the number of sides and the angles in a polygon.