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Mastering the Geometry Puzzle of Inscribed Circles

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Chapter 1: Understanding the Puzzle

Have you ever encountered a geometry puzzle involving four smaller circles inscribed within a larger circle? Many Gothic cathedrals feature stained glass windows that include a design of congruent circles encircled by a larger circle. In the illustration below, you can see that there are four smaller circles.

Inscribed circles in a larger circle

Now, the question arises: What is the ratio of the combined areas of these four smaller circles to the area of the larger circle? Here’s a helpful hint: let’s assume the radius of the smaller circles is 1. Take a moment to pause, grab your pen and paper, and try to solve it before continuing to the solution!

Section 1.1: The Calculation Process

To simplify our calculations, we will consider the radius of each smaller circle to be 1. The centers of these circles form a square with a side length of 2.

Square formed by the centers of the smaller circles

It’s important to note that the diagonal of the square serves as the hypotenuse of a right triangle that is half the size of the square. According to the Pythagorean Theorem, we find that:

(diagonal length)² = 2² + 2², thus the diagonal length = 2√2.

Next, let’s examine the diameter of the larger circle. The total diameter consists of the diagonal plus the two radii of the smaller circles:

diameter = 1 + 2√2 + 1 = 2 + 2√2.

Once we have the diameter, we can easily determine the radius and subsequently the area of the larger circle.

Area calculation of the larger circle

The area of all four smaller circles can be calculated as 4π since each circle has a radius of 1.

Now, let’s bring together our findings to calculate the desired ratio.

Area ratio of smaller circles to the larger circle

And there you have it—our answer! Isn’t it fascinating?

What did you think of this problem? I would love to hear your thoughts in the comments below!

Chapter 2: Video Insights

To further enhance your understanding, check out these videos that delve into similar geometry challenges.

Discover the easiest method to tackle this geometry puzzle and sharpen your problem-solving skills!

Uncover the surprising answer to this geometry puzzle that challenges your perception!

Thank you for engaging with this content. If you found it informative, please consider sharing it with friends who enjoy math puzzles. Your feedback is invaluable, and I appreciate your support!

Geometry puzzle appreciation

If you feel inclined to support my writing, consider buying me a coffee. Your generosity helps sustain my academic and writing endeavors. Happy solving!

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