Drawing a Circle Inscribed in a Hexagon Using Python

Drawing a circle inscribed in a hexagon using Python involves understanding the geometric relationship between a hexagon and its inscribed circle. A hexagon has six equal sides and six equal angles, and when a circle is inscribed inside it, the circle touches each of the hexagon’s sides at exactly one point. This point of tangency is where the radius of the circle is perpendicular to the side of the hexagon.

To draw such a figure using Python, we can leverage libraries like matplotlib for plotting and numpy for mathematical operations. The steps to achieve this are as follows:

1.‌Calculate the Radius of the Inscribed Circle‌: For a hexagon with side length s, the radius r of the inscribed circle can be calculated using the formula $r=s\cdot \frac{\sqrt{3}}{2}$. This formula is derived from the geometric properties of a hexagon and its inscribed circle.

2.‌Define the Hexagon’s Coordinates‌: To draw the hexagon, we need to calculate the coordinates of its vertices. This can be done by placing the hexagon in a convenient position in the coordinate plane, such as centering it at the origin, and then calculating the coordinates of each vertex based on the side length and the angles between the sides.

3.‌Plot the Hexagon and the Inscribed Circle‌: Using matplotlib, we can plot the hexagon by drawing lines between its vertices and plot the inscribed circle using its center and radius.

Here’s a simple Python script that demonstrates how to draw a hexagon with an inscribed circle:

pythonCopy Code
import numpy as np
import matplotlib.pyplot as plt

def draw_hexagon_with_inscribed_circle(side_length):
    # Calculate the radius of the inscribed circle
    radius = side_length * np.sqrt(3) / 2
    
    # Define the center of the hexagon (and the circle)
    center = (0, 0)
    
    # Calculate the coordinates of the hexagon's vertices
    num_vertices = 6
    angles = np.linspace(0, 2 * np.pi, num_vertices + 1)[:-1]
    vertices = np.array([center + radius * np.array([np.cos(angle), np.sin(angle)]) for angle in angles])
    
    # Plot the hexagon
    plt.plot(vertices[:, 0], vertices[:, 1], 'b-')
    
    # Plot the inscribed circle
    circle = plt.Circle(center, radius, color='r', fill=False)
    plt.gca().add_artist(circle)
    
    plt.axis('equal')
    plt.show()

# Example usage
draw_hexagon_with_inscribed_circle(5)

This script calculates the radius of the inscribed circle, defines the coordinates of the hexagon’s vertices, and then plots both the hexagon and the inscribed circle using matplotlib. By running this script, you will see a visualization of a hexagon with an inscribed circle, demonstrating the geometric relationship between the two shapes.

[tags]
Python, Geometry, Hexagon, Inscribed Circle, Matplotlib, Numpy