Calculating the Area of Overlap Between Two Overlapping Circles: A Comprehensive Guide

When dealing with the area of overlap between two circles that intersect, also known as the lens-shaped area or the intersection area, it requires a detailed understanding of circle geometry and the application of specific formulas. In this guide, we will explore the step-by-step process to find the area of overlap between two circles with a radius of 4 cm, where the center of each circle lies on the circumference of the other.

Geometry and Given Values

To start, let us establish the given values:

Radius of each circle, r: 4 cm Distance between the centers of the two circles, d: 4 cm (equal to the radius of each circle)

With these values, we can now apply the formula for the area of overlap between two circles.

Formula for the Area of Overlap

The area of overlap A between two circles of radius r with a distance d between their centers is given by:

A 2r2 cos-1 (d / (2r)) - (1/2)d sqrt(4r2 - d2)

Substituting the given values into the formula:

A 2(42) cos-1 (4 / (2*4)) - (1/2)4 sqrt(4(42) - 42)

Calculating the Components

Let's break down the steps to simplify and solve for the area of overlap:

Calculate d / (2r):

d / (2r) 4 / (2*4) 1/2

Find cos-1(1/2):

cos-1(1/2) π/3

Substitute the values back into the formula:

A 2(16) (π/3) - (1/2)4 sqrt(4(16) - 16)

A (32π/3) - 2 sqrt(64 - 16)

A (32π/3) - 2 sqrt(48)

A (32π/3) - 2 (4√3)

A (32π/3) - 8√3

Therefore, the area of the shape common to the two overlapping circles is:

A (32π/3) - 8√3 cm2

Alternative Approach: Segment and Triangle Breakdown

Another way to approach this problem is by breaking the area into simpler geometric shapes. The area can be divided into a one-third segment of one of the circles and two little wedges. Each little wedge will have the same area as a one-sixth segment of a circle minus the area of an equilateral triangle with sides equal to the radius of our circles.

Area of a one-third segment: (πr2 / 3) Area of a one-sixth segment: (πr2 / 6) Area of an equilateral triangle with sides r: (r2√3 / 4)

The total area of the two little wedges will be:

(1/3)r2π 2((1/6)r2π - (1/4)r2√3) r2(2π/3 - √3/2)

Substituting r 4 cm into the formula:

16(2π/3 - √3/2) cm2 ≈ 19.65 cm2

Conclusion

This comprehensive guide offers a detailed method to calculate the area of intersection between two overlapping circles of radius 4 cm. By applying the formula for the area of overlap and breaking the area into simpler geometric shapes, we can accurately determine the size of the lens-shaped area formed by the intersection of the circles.