Introduction to Square Area Calculation in a Circle

The geometric relationship between a square inscribed within a circle is a fascinating topic in mathematics. This article aims to explore the process of finding the area of the largest possible square that can fit inside a circle with a given area of 36π cm2. This knowledge is not only foundational in geometry but also useful in various practical applications, from architecture to design.

Area of a Circle with Radius 6 cm

To begin, let us start by calculating the radius of the circle from its area. The area of a circle is given by the formula:

A πr2

Given that A 36π cm2, solving for r involves a simple algebraic manipulation:

36π πr2

Divide both sides by π:

36 r2

Take the square root:

r 6 cm

This gives us the radius. Now, the diameter of the circle can be calculated as:

d 2r 2 × 6 12 cm

Diagonal of the Largest Square

The largest square that can fit inside a circle has its diagonal equal to the diameter of the circle. Let’s denote the side of the square as s. The relationship between the diagonal and the side of the square is given by:

Diagonal (d) s√2

Given that the diagonal of the square is the diameter of the circle, we have:

d s√2 12 cm

Thus, solving for s gives us:

s 12 / √2 12 × (√2 / 2) 6√2 cm

Area of the Square

Finally, the area of the square, As, can be calculated using the formula:

As s2

Substituting s 6√2 cm, we get:

As (6√2)2 36 × 2 72 cm2

Hence, the area of the largest possible square that can fit inside the circle is 72 cm2.

Geometric Explanation: Base-Angle Theorem and Congruent Triangles

The geometric relationship between the square and the circle can also be understood through the properties of congruent triangles. In the circle, the diameter serves as the diagonal of the inscribed square. Since the diameter forms a 90-degree angle with any radius at the point where they intersect, each triangle formed within the square is a 45-45-90 right triangle.

Given that the radius (r) is 6 cm, the hypotenuse (diagonal of the square) is 12 cm. Using the properties of the 45-45-90 triangle, the side length of the square can be determined as:

Side (s) 6√2 cm

The area can then be calculated as:

As 4 × (62 / 2) 4 × (36 / 2) 72 cm2

This confirms the area of the largest possible square inscribed within the circle.

Conclusion

Understanding the geometric relationships and calculations for determining the area of the largest square inscribed within a circle is a crucial concept in geometry. This not only enhances mathematical skills but also aids in practical applications where such geometric properties are utilized. The area of 72 cm2 for the largest square in a circle with an area of 36π cm2 is a clear and concise example of these geometric principles in action.