Finding the Ratio of the Area of a Circular Sector to the Area of a Square

In geometry, there are several interesting problems involving the circular sector and the inscribed circle within a square. One such intriguing puzzle is to find the ratio of the area of a red sector in a circle inscribed in a square to the total area of the square. Understanding this relationship can help in numerous applications, ranging from architecture to design and engineering.

Understanding the Geometry

Consider a square with each side of length LL. The diameter of the circle inscribed in the square is equivalent to the side length of the square, so the radius RR of the circle is L/2L/2.

Calculating the Area of the Red Sector

The red sector of the circle has a central angle of 18°. The area of a circular sector is given by:

A1360θR2Afrac{1}{360}theta R^2

Plugging in the values:

A18360πL22πL28Afrac{18}{360}pileft(frac{L}{2}right)^2pifrac{L^2}{8}

The area of the square is L22L^2. Thus, the ratio of the area of the red sector to the area of the square is:

Ratioπ:8Ratio pi:8

Simplifying the ratio:

Ratio1:25.46Ratio 1:25.46

Alternative Calculation Methods

Another method to solve the problem involves expressing the area of the red sector and the square in terms of the side length of the square, denoted as xx.

The area of the square is x22x^2. The area of the inscribed circle is 2228x2frac{22}{28}x^2. The area of the red sector is:

A183602228x20.039285714x2Afrac{18}{360}frac{22}{28}x^20.039285714x^2

The required ratio is:

Ratio0.039285714139:1000Ratio frac{0.039285714}{1}39:1000

Conclusion

Understanding the geometric relationship and the areas involved in a circular sector inscribed in a square provides a fundamental insight into geometric ratios and proportions. These concepts are not only crucial in theoretical geometry but are also applicable in practical scenarios, such as designing and construction. The ratio of the area of the red sector to the area of the square serves as a useful measure