#### 65**Question:** A circle has a radius of \( 5 \) units, and a square is inscribed within it. What is the area of the square? Express your answer in terms of \(\pi\).
["### How to Find the Area of a Square Inscribed in a Circle with Radius 5 Units", "#### Understanding the Geometry: Circle and Inscribed Square", "When a square is inscribed inside a circle, all four vertices of the square touch the circumference of the circle. This means the circle’s diameter serves as the diagonal of the square. Given the circle’s radius is 5 units, the diameter — and thus the square’s diagonal — is:", "[\n\ ext{Diameter} = 2 \ imes \ ext{radius} = 2 \ imes 5 = 10 \ ext{ units}\n]", "#### Relating the Diagonal to the Square’s Side Length", "Let the side length of the inscribed square be ( s ). For a square, the relationship between the side length and the diagonal is governed by the Pythagorean theorem:", "[\n\ ext{Diagonal} = s\sqrt{2}\n]", "Since the diagonal equals 10 units, we set up the equation:", "[\ns\sqrt{2} = 10\n]", "Solving for ( s ):", "[\ns = \frac{10}{\sqrt{2}} = 5\sqrt{2}\n]", "#### Calculating the Area of the Square", "The area ( A ) of the square is the square of its side length:", "[\nA = s^2 = (5\sqrt{2})^2 = 25 \ imes 2 = 50\n]", "Although the circle’s radius involves ( \pi ), the area of the inscribed square is a simple geometric value — but let’s clarify the requested expression. While the answer is a numerical value (50 square units), the problem asks to “express your answer in terms of ( \pi ).” However, since the area does not directly involve ( \pi ), this reflects a subtle design nuance — the circle’s presence defines the square’s size, but the square’s area is constant regardless of ( \pi ).", "If interpreted differently, some variations might express side length with ( \pi ), but here the final area is:", "[\n\boxed{50}\n]", "Note: In full geometric problems involving circles and inscribed shapes, although ( \pi ) influences radial measures, areas depending on inscribed polygons are numerical and independent of ( \pi ), unless redefined via angular parameters. This solution reflects standard Euclidean geometry.", "---", "SEO Meta Description:\nLearn how to calculate the area of a square inscribed in a circle with radius 5 units. Step-by-step geometric solution defining the diagonal as diameter, finding side length, and computing area—clear, accurate, and optimized for SEO with keywords like “inscribed square in circle area,” geometric problems, and circle-inscribed shapes.", "---", "Key SEO Elements:\n- Target keyword: “area of square inscribed in circle radius 5”\n- Schema-relevant headings: Question, Explanation, Step-by-step, Summary\n- Semantic-rich phrases: “circle radius 5”, “inscribed square diagonal”, “side length square”\n- Internal linking hints: “Learn how to find inscribed square areas,” “geometric circle problems”\n- Readability and clarity for user intent and search intent alignment."]
