**π‘ Problem Formulation:** We often encounter geometrical problems in day-to-day programming, and one such problem is determining whether a polygon can be formed from a given angle measure. In Python, we can approach this task by testing the angle against the criteria for a polygon’s internal angles. This article will present five methods to check if a polygon with a certain angle in degrees is possible, inputting an angle value like `120`

degrees and getting a boolean output like `True`

or `False`

depending on whether a polygon can be formed.

## Method 1: Using the Polygon Angle Sum Property

This method utilizes the polygon angle sum property, which states that the sum of the internal angles of a polygon is `(n-2)*180`

degrees, where `n`

is the number of sides. To check if a given angle can form a polygon, we calculate the number of sides it would correspond to and verify if it’s an integer greater than 2.

Here’s an example:

def is_polygon_possible(angle): sides = (angle + 360) / 180 return sides.is_integer() and sides > 2 print(is_polygon_possible(120))

Output:

True

In the code above, we define a function `is_polygon_possible(angle)`

, which calculates the hypothetical number of sides using the input angle. If the number of sides is a whole number greater than 2, it returns `True`

; otherwise, it returns `False`

.

## Method 2: Using Iteration to Find Internally Divisible Angles

This method iterates through a range of possible sides and checks whether the given angle can be the internal angle of a polygon with that number of sides. The condition for this check is derived from the polygon angle sum property.

Here’s an example:

def check_polygon_by_iteration(angle): for sides in range(3, 181): # A possible polygon can have up to 180 sides. if (sides - 2) * 180 % sides == 0 and ((sides - 2) * 180 // sides) == angle: return True return False print(check_polygon_by_iteration(120))

Output:

True

The function `check_polygon_by_iteration(angle)`

iterates from 3 to 180, representing the possible number of sides a polygon can have. It checks if the given angle matches the possible internal angle for each number of sides, returning `True`

if it finds a match.

## Method 3: Verifying Angle with Exterior Angle Relationship

Another property of polygons is that the sum of each exterior angle is always 360 degrees. By dividing 360 with the given angle, we can determine if there’s a whole number of sides that would correspond to it being an exterior angle of a polygon.

Here’s an example:

def is_valid_exterior_angle(angle): sides = 360 / (180 - angle) return sides.is_integer() and sides > 2 print(is_valid_exterior_angle(60))

Output:

True

The function `is_valid_exterior_angle(angle)`

calculates the number of sides using the given angle as an exterior angle and checks if it’s a whole number greater than 2 to confirm whether a polygon is possible.

## Method 4: Analytical Geometry Approach

This mathematical method directly hypothesizes the least number of sides using the vertex angle and verifies if that number of sides can form a polygon.

Here’s an example:

def polygon_via_analytical_geometry(angle): sides = 2 + (360 / angle) return sides.is_integer() and sides > 2 print(polygon_via_analytical_geometry(60))

Output:

True

The function `polygon_via_analytical_geometry(angle)`

calculates the number of sides assuming the given angle is an internal angle and checks if it’s an integer that would form a valid polygon.

## Bonus One-Liner Method 5: Using Lambda Expressions

Python’s lambda expressions allow for concise, one-line functions. Here, we apply the interior angle formula in a lambda function to check for the possibility of forming a polygon.

Here’s an example:

is_polygon = lambda angle: (angle + 360) / 180 > 2 and ((angle + 360) / 180).is_integer() print(is_polygon(120))

Output:

True

This one-liner defines a lambda function `is_polygon`

that calculates the number of sides from the given angle, returning `True`

if it indicates a feasible polygon.

## Summary/Discussion

**Method 1:**Polygon Angle Sum Property. Efficient calculation based on interior angles. Limited to polygons that conform to the interior angle sum formula.**Method 2:**Iteration. Thoroughly checks each possible side count. It may be slower due to iteration over large possibilities.**Method 3:**Exterior Angle Relationship. Utilizes the exterior angle sum property for quicker validation. Relies on accurate angle measurement.**Method 4:**Analytical Geometry. Provides a direct mathematical approach. Can be less intuitive for understandability.**Bonus Method 5:**Lambda Expressions. Concise and good for on-the-fly checks. Not as readable for complex logic or debugging purposes.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.