Maximizing the Area of the Largest Rectangle Inscribed in a Circle

Introduction

Geometry is an essential branch of mathematics that often reveals surprising properties and relationships. One interesting problem is determining the largest rectangle that can be inscribed within a circle of a given radius. This article will explore the geometry of the situation and provide a step-by-step solution to find the maximum area of this inscribed rectangle.

Understanding the Geometry

The largest rectangle that can be inscribed in a circle is a square. This is due to the fact that for a given perimeter, a square has the maximum area among all rectangles. The principle of maximizing area in such a geometric context can be seen in various real-life applications, from designing efficient layouts to optimizing space in architecture.

Radius and Diameter

For a circle with a radius of 1, the diameter is calculated as:

Diameter 2 × radius 2 × 1 2

Square Inscription

When a square is inscribed in a circle, its diagonal equals the diameter of the circle. Let the side length of the square be (s). The diagonal (d) of the square can be calculated using the Pythagorean theorem:

d s(sqrt{2})

Setting the diagonal equal to the diameter, we have:

ssqrt{2} 2

Solving for the side length (s):

s (frac{2}{sqrt{2}}) (sqrt{2})

Calculating the Area

The area (A) of the square is given by:

A (s^2) ( (sqrt{2})^2) 2

Therefore, the area of the largest rectangle that can be inscribed in a circle of radius 1 is 2 square units.

General Case

A similar reasoning can be applied to a circle with a general radius (r). The diagonal of the inscribed square will be (2r), and using the same steps, we find that the side length (s) is:

s (frac{2r}{sqrt{2}}) (sqrt{2}r)

The area (A) of the square becomes:

A ((sqrt{2}r)^2) 2r^2

This confirms that the area of the largest rectangle inscribed in a circle of radius 1 is 2 square units.

Conclusion

The problem of the largest rectangle inscribed in a circle demonstrates the elegance of geometric principles. By understanding the relationship between the square and the circle, we can derive a formula that applies to any circle, regardless of its size. This knowledge can be useful in various fields, including engineering, architecture, and design.