A square is circumscribed about a circle of 7 -foot radius, as shown below. What is the area of the square, in square feet? \(\begin{array}{ll}{\mathbf{F} .} & {49} \\ {\mathbf{G} .} & {56} \\\ {\mathbf{H} .} & {98} \\ {\mathbf{J} .} & {49 \pi} \\ {\mathbf{K} .} & {196}\end{array}\)

Understanding the concept of a circumscribed square involves recognizing the spatial relationship between a square and a circle. In geometry, when a square is circumscribed around a circle, each side of the square touches the circle at exactly one point. This means that the circle fits perfectly inside the square, and the diameter of the circle coincides with the diagonal of the square.

To calculate the area of such a square, we first need the length of the diagonal, which can be found by doubling the radius of the circle. With the circle's diameter known, we can calculate the side length of the square using the Pythagorean Theorem—as the diagonal forms a right triangle with any two adjacent sides. Once the side length is determined, the area can be found by squaring this length. It is a process that requires methodical application of basic geometry principles.

The Pythagorean Theorem is a cornerstone of geometry, particularly in right-angled triangles. It states that for such a triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed in a formula, it is written as \(c^2 = a^2 + b^2\), where c is the hypotenuse, and a and b are the other sides.

In problems involving circumscribed squares, the Pythagorean Theorem allows us to find the square's side length by using the diagonal as the hypotenuse. This application underscores the theorem's utility in finding unknown dimensions in a variety of geometric configurations, and why it's an essential tool in a student's math arsenal.

Geometry is all about the properties and relations of points, lines, surfaces, and solids in space. When dealing with a problem that includes both a circle and a square, it is vital to understand the properties of these shapes. For instance, the fact that the diagonal of the square equals the diameter of the circumscribed circle is a reflection of the symmetry and proportional relationships inherent in geometric figures.

An appreciation of geometry's fundamental concepts, like congruence, similarity, and the characteristics of polygons and circles, allows students to visualize and solve problems more effectively. Visual aids, such as drawing the problem, can be particularly helpful for students to grasp the spatial relationships between different shapes.

Calculating the diameter of a circle is often the first step in solving geometry problems involving circles. The diameter is the longest distance from one point on the circumference of the circle to another, passing through the center. It is twice the length of the radius. The formula to find the diameter is simple: \(D = 2r\), where D stands for diameter and r for radius.

This calculation is crucial when dealing with circumscribed figures, as the diameter becomes a key measurement that often relates to other elements of the figure, such as the diagonal of a circumscribed polygon. By doubling the given radius, we quickly move from understanding a component of the circle to integrating it into the broader context of the problem at hand.