Calculating the Area of the Shaded Square: A Geometry Puzzle

According to the image provided in , the numbers were filled in the order green, orange, red, and pink. This geometry problem involves calculating the area of a shaded square, which requires a combination of various geometric principles.

Understanding the Geometry Problem

The geometry problem can be summarized as follows: Calculate the area of the shaded square, given the area of the box, the areas of the circles, and the small missing portions. The numbers provided are green, orange, red, and pink, which help in breaking down the problem into smaller, manageable parts.

Step-by-Step Solution

The formula to find the required area of the shaded square can be expressed as:

Required Area Area of Box - Area of Circles/2 - Area of the Small Missing Portion

This can be mathematically written as:

Required Area 200 cm2 - π(252/2) - X

To find the value of X, we need to break down the problem further. X represents the area of the small missing portion, which consists of the area of the isosceles triangle, the area covered by the arc, and the small remaining area.

Calculating X

The angle y is calculated as follows:

angle y 180 - 226.565 126.87 degrees

The area of the isosceles triangle (W) can be calculated using the formula:

W 1/2 × R2 × Siny 0.5 × 7.9992 × sin(126.87) 10 cm2

The area covered by the arc (Y) can be calculated using the formula:

Y π × (252) × (126.87/360) 27.679 cm2

The area of the box (Z) can be calculated using the formula:

Z Area of Box - Area of Circles/8 5.365 cm2

The value of X is then calculated as:

X Area of Triangle - Z - Y 0.5 - 5.365 - 27.679 1.956 cm2

Final Calculation

Now, let us calculate the required area:

Required Area 200 cm2 - π(252/2) - 1.956 21.46 - 1.956 19.504 cm2

Therefore, the required area of the shaded square is 19.504 cm2.

Conclusion

This problem showcases the application of various geometric principles, including the calculation of areas of circles, triangles, and arcs. It also emphasizes the importance of breaking down complex problems into smaller, understandable parts.

Understanding these principles is crucial for solving similar geometry problems and can be applied in various real-world scenarios, such as in engineering, architecture, and design.