Calculating the Area Bounded by Tangents and Circles: A Comprehensive Tutorial

In this article, we will explore the concept of how to calculate the area bounded by two circles and a common tangent. This involves geometric principles and calculus to determine the exact area of the region. This type of problem is not only important in geometry but also useful in various scientific and engineering applications.

Understanding the Geometry of the Problem

Consider two circles whose centers are denoted as (O) and (P). The circles touch each other externally, and a common tangent (XY) is drawn. The radii of the circles are given as (12 cm) and (4 cm), respectively. The task is to find the area of the region bounded by the circles and the tangent line (XY).

Step 1: Determining the Length of the Tangent Segment (XY)

Since the circles touch externally, the distance (OP 12 4 16) cm. The tangent segment (XY) can be calculated using the Pythagorean theorem in the right triangle formed by the radii and the tangent. Here, the altitude from the point of tangency is the difference in the radii (12 - 4 8) cm.

[XY sqrt{16^2 - 8^2} sqrt{256 - 64} sqrt{192} 8sqrt{3}text{ cm}]

Step 2: Finding the Area of the Trapezium (OxyA)

Let's consider the trapezium (OxyA). The area of the trapezium can be calculated using the formula for the area of a trapezium. Here, the parallel sides are (16) cm (the distance between the centers) and (8sqrt{3}) cm. The height of the trapezium is the same as the difference in the radii, which is (8) cm.

[A_{trap} frac{1}{2} times (16 8sqrt{3}) times 8 64sqrt{3}text{ cm}^2]

Step 3: Calculating the Area of the Sectors

Next, we need to find the areas of the sectors of the circles. The angles subtended by the sectors at the centers are determined by the geometric relationships involved. For the sector of the larger circle with center (O), the angle is (60°), and for the sector of the smaller circle with center (P), the angle is (120°).

The area of the sector of the larger circle is:

[A_{sector1} frac{60pi times 12^2}{360} 24pitext{ cm}^2]

The area of the sector of the smaller circle is:

[A_{sector2} frac{120pi times 4^2}{360} frac{16pi}{3}text{ cm}^2]

Step 4: Finding the Area Between the Circles and Tangents

The area between the circles and the tangents is essentially the area of the trapezium minus the sum of the areas of the sectors.

[A_{between} 64sqrt{3} - left(24pi frac{16pi}{3}right) 64sqrt{3} - frac{88pi}{3}text{ cm}^2]

Step 5: Calculating the Total Area Bounded by Circles and Tangents

The total area bounded by the circles and the tangents is the sum of the areas of the individual sectors of the circles minus the area between the circles and the tangents.

[A_{total} pi times 12^2 pi times 4^2 - left(64sqrt{3} - frac{88pi}{3}right) 144pi 16pi - 64sqrt{3} frac{88pi}{3}]

[A_{total} 169pi - 64sqrt{3}text{ cm}^2]

Conclusion

The area bounded by the two circles and the tangent (XY) is approximately:

(169pi - 64sqrt{3}) which evaluates to approximately (514.4) cm2.

This calculation involves a careful understanding of geometric principles, the application of the Pythagorean theorem, and trigonometric identities. The problem demonstrates the interplay between geometry, algebra, and trigonometry in solving complex geometric problems.