Using Python to Compute the Shaded Area of a Sixth‑Grade Geometry Problem

The author encountered a sixth‑grade mathematics question that asked for the area of a shaded region, found it challenging to solve analytically, and therefore turned to Python to compute the area.

Idea Introduction : First, the picture is reproduced on a coordinate plane using Python, defining the slanted line and the semicircle as functions. Then, the rectangle is uniformly filled with a large number of points (conceptually "beans"); the proportion of points that fall inside the shaded region is used to estimate its area.

Source Code Design :

1. Code for drawing the figure.

2. Code for the calculation, where the parameter * represents the total number of sampled points, side indicates whether points on the border are counted (theoretically should be True), and t controls the number of iterations – larger t yields longer runtime but higher precision.

Conclusion : With t = 100 the estimated shaded area is 1.268; with t = 1000 it becomes 1.253, which is very close to the correct answer of 1.252. The approach shows that any irregular shape whose boundary can be expressed as a function can be handled similarly. Three solution methods are presented, the third being particularly intricate.