Finding the Radius of a Circle Inscribed in a Square with a Given Perimeter

In this article, we will explore a geometrical problem that frequently arises in both academic and real-world applications: determining the radius of a circle inscribed within a square, given the perimeter of the square. This problem is not only interesting in its own right but also provides valuable insights into the relationships between different geometric figures.

Understanding the Problem

The problem states that a circle is inscribed within a square, meaning the circle is tangent to all four sides of the square. The distance around the square is given as 36 cm, which is the perimeter of the square. We need to find the radius of the circle under these conditions.

Step-by-Step Solution

Step 1: Determine the Side Length of the Square

First, we need to find the side length of the square. The perimeter (P) of a square is given by the formula:

P 4s

where s is the side length of the square. Given that the perimeter is 36 cm, we can set up the equation:

4s 36

Dividing both sides by 4, we get:

s frac{36}{4} 9 , text{cm}

Step 2: Determine the Diameter of the Circle

Since the circle is inscribed in the square, the diameter of the circle is equal to the side length of the square. Therefore:

Diameter s 9 , text{cm}

Step 3: Calculate the Radius of the Circle

The radius (r) of the circle is half of the diameter. Therefore, we have:

r frac{text{Diameter}}{2} frac{9}{2} 4.5 , text{cm}

Thus, the radius of the circle is 4.5 cm.

Alternative Approaches and Further Exploration

The solution to this problem can be approached in multiple ways, each offering a unique perspective on the interplay between the square and the circle. Here are a few alternative methods:

Method 1: Direct Calculation

We can directly compute the side length of the square from the perimeter, and then use the relationship between the diameter and radius of the circle:

Length of one side of the square frac{36}{4} 9 , text{cm}

Radius of the circle frac{9}{2} 4.5 , text{cm}

Method 2: Circumference and Area

Consider the square with side length 9 cm. The radius of the circle, which is inscribed, can be determined using the formula for the radius of a circle inscribed in a square:

Radius frac{text{Side length}}{2} frac{9}{2} 4.5 , text{cm}

Method 3: Analytic Geometry Approach

In an analytic geometry framework, the side length of the square is determined by the given perimeter, and the radius of the inscribed circle can be found using linear algebra:

Side length frac{36}{4} 9 , text{cm}

Radius frac{9}{2} 4.5 , text{cm}

Conclusion

In summary, the radius of a circle inscribed in a square with a given perimeter can be determined by first finding the side length of the square and then halving it to get the radius of the circle. The solution to our problem is: the radius of the circle is 4.5 cm.

This problem not only helps us understand the relationship between geometric shapes but also reinforces the importance of mathematical problem-solving skills in various applications.