# 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.

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.

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.

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.

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!

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