Question

1. what is the area of a square inscribed in a circle of radius \\(r\\)?

\\(2r^2\\)
\\(2r^3\\)
\\(2\pi r\\)
\\(4r^2\\)

Explanation:

Identify the geometric relationship

The square is inscribed in a circle of radius \(r\).
The diagonal of the inscribed square is equal to the diameter of the circle.
The diameter \(d\) of the circle is:

$$d = 2r$$

Relate diagonal to side length

Let \(s\) represent the side length of the square.
Using the Pythagorean theorem for the right triangle formed by two sides and the diagonal:

$$s^2 + s^2 = d^2$$

$$2s^2 = (2r)^2$$

Solve for the area of the square

The area \(A\) of a square is given by \(s^2\).
Simplify the equation to find \(s^2\):

$$2s^2 = 4r^2$$

$$s^2 = 2r^2$$

Thus, the area of the square is:

$$A = 2r^2$$

Answer:

• (A) \(2r^2\) (Correct answer)
• (B) \(2r^3\)
• (C) \(2\pi r\)
• (D) \(4r^2\)