A circle is inscribed in a square, which is circumscribed by...

A circle is inscribed in a square, which is circumscribed by another circle. If the diagonal of the square is $2 x,$ find the ratio of the area of the large circle to the area of the small circle.

Key Concepts

Area of a Circle

Calculating the area of a circle depends on the radius, with the area given by the formula A = ?r². Understanding how the radius relates to other dimensions in the problem, such as the square’s side length or diagonal, allows one to compare the areas of the inscribed and circumscribed circles.

Ratio of Areas

This concept involves comparing the sizes of two geometric shapes by their areas. In the problem, after determining the radii of the inscribed and circumscribed circles, the ratio of their areas is found by dividing one area by the other. The ratio is particularly straightforward when the dimensions of the circles are directly related through the geometry of the square.

Inscribed and Circumscribed Figures

This concept involves understanding the spatial relationship where one shape is exactly contained within another. In the context of the problem, a circle is inscribed in a square, meaning the circle touches the square's sides at exactly one point per side, while another circle is circumscribed about the square, meaning the square is entirely contained within the circle and all its vertices lie on the circle. This concept is crucial for relating the dimensions of the square to those of the circles.

Relationship Between Square Diagonals and Side Length

A square's diagonal is related to its side length by the factor of ?2, since the diagonal equals the side multiplied by ?2. This relationship is key to converting the given diagonal measurement into the side length of the square, which is then used to determine the dimensions (diameter and radius) of the inscribed circle.