Borek said:Try to express surface of the inscribed 4-sided polygon as a function of the length of its sides. General case will be more difficult, but for a specific case of rectangle it should be relatively easy to express lengths as a function of one variable.
Once you have that, find maximum.
The proof involves using the properties of a circle and a square to show that the area of the square is equal to the area of the circle. This can be done using geometrical constructions and mathematical equations.
The formula is A = (π/4) * r^2, where A is the area of the square and r is the radius of the circle. This formula is derived from the fact that the diagonal of the square is equal to the diameter of the circle, and the area of a square is calculated by multiplying the length of its side by itself.
This is because a square is the only quadrilateral that can be inscribed in a circle and have all of its vertices touching the circumference of the circle. This results in the square having a larger area compared to any other quadrilateral inscribed in the same circle.
Yes, it is possible to visually see that a square inscribed in a circle has the maximum area by using geometrical constructions and comparing the areas of different shapes inscribed in the same circle. However, using the formula will provide a more accurate and efficient method of calculation.
This concept is useful in many areas such as architecture, engineering, and mathematics. It can be applied in designing structures with maximum stability and efficiency, optimizing the use of space in various applications, and solving various mathematical problems involving circles and squares.