Calculating the Area of Intersection Between Two Circles

In geometry, the intersection of two circles can present a fascinating challenge in terms of area calculation. This article explores the method to find the area of the intersecting region of two identical circles each with a radius of 4 cm, where each circle passes through the center of the other. The steps include determining the distance between the centers of the circles, applying a specific formula, and identifying the appropriate geometric shapes.

Geometric Configuration and Preliminary Calculations

Given two identical circles, each with a radius of r 4 cm, and the condition that each circle's circumference passes through the center of the other, the distance between the centers of the circles, d, is equal to the radius:

d r 4 cm

Formula for the Area of Intersection

The area A of the intersection of two circles of radius r that are d units apart is given by the formula:

A 2r^2 cos^{-1} left( frac{d}{2r} right) - frac{d}{2} sqrt{4r^2 - d^2}

Substituting the Values

Given r 4 cm and d 4 cm, we can proceed with the following calculations:

frac{d}{2r} frac{4}{2 times 4} frac{4}{8} frac{1}{2} cos^{-1} left( frac{1}{2} right) frac{pi}{3}

Calculation of the Area

Using the area formula:

A 2 times 4^2 times frac{pi}{3} - frac{4}{2} sqrt{4 times 4^2 - 4^2}

Calculate the first term:

2 times 16 times frac{pi}{3} frac{32pi}{3}

Calculate the second term:

2 times sqrt{4 times 16 - 16} 2 times sqrt{64 - 16} 2 times sqrt{48} 2 times 4sqrt{3} 8sqrt{3}

Therefore, the area of the intersection is:

A frac{32pi}{3} - 8sqrt{3}

The final result is:

The area of the intersecting region of the two circles is: A frac{32pi}{3} - 8sqrt{3} , text{cm}^2

Alternative Method of Calculation

An alternative method involves calculating the area using segments and equilateral triangles. The overlapping area is composed of:

2 equilateral triangles 4 segments

A segment area is given by:

text{Segment area} text{sector area} - text{equilateral triangle area} frac{1}{2} r^2 Theta - frac{sqrt{3}}{4} r^2

The total area of the intersection using this method is:

A 2 times text{equilateral triangle area} 4 times text{segment area}

This method ultimately leads to the same result as the first method, confirming the accuracy of the calculation.

Final Result

The area of the intersecting region of the two circles is:

A frac{32pi}{3} - 8sqrt{3} , text{cm}^2

Conclusion

Understanding the area of intersection between two circles can be crucial in various fields, including engineering, architecture, and design. The steps and methods discussed here provide a clear and systematic approach to solving such problems.