Question

A square is inscribed in a circle. The area of the square is what percent of the area of the circle? (Disregard the percent symbol when gridding your answer.)

Short answer

The area of the inscribed square is approximately \(63.66\%\) of the area of the circle.

Step-by-step solution

1. Circle and Square Relationship

First, let's examine the relationship between the circle and the inscribed square. The vertices of the square touch the circumference of the circle, so the diagonal of the square is equal to the diameter of the circle.

2. Square Side Length

Let's denote the side length of the square as \(s\), and the diagonal of the square as \(d\). We can use the Pythagorean theorem to relate the diagonal with the sides of the square. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the other two sides. The diagonal of the square is the hypotenuse of a right triangle formed by two sides of the square: \[d^2 = s^2 + s^2\]

3. Circle's Radius Relationship

Now, we need to find the relationship between the diagonal of the square (\(d\)) and the radius of the circle (\(r\)). Since the diagonal is equal to the diameter of the circle, then: \[d = 2r\]

4. Side Length in terms of Radius

Now we can use the equation for the diagonal to find the side length (\(s\)) in terms of the circle's radius (\(r\)). We have these two equations: - \(d^2 = 2s^2\) - \(d = 2r\) We can substitute the second equation into the first equation to get: \[(2r)^2 = 2s^2\] Which gives us: \[4r^2 = 2s^2\] Now, we can solve for \(s\): \[s^2 = 2r^2\] So, \(s = \sqrt{2r^2}\)

5. Compute Areas of Square and Circle

Now that we have the side length of the square in terms of the circle's radius, we can compute the areas of both shapes: - Area of the square (\(A_s\)) : \(A_s = s^2 = 2r^2\) - Area of the circle (\(A_c\)) : \(A_c = \pi r^2\)

6. Compute the Ratio as a Percentage

To find the area of the square as a percentage of the area of the circle, we will compute the ratio of their areas and multiply by 100: \[\frac{A_s}{A_c} \times 100 = \frac{2r^2}{\pi r^2} \times 100\] Notice that \(r^2\) cancels out from the numerator and denominator, leaving: \[\frac{2}{\pi} \times 100\] Now, compute the value and round to the nearest integer: \[\frac{2}{\pi} \times 100 \approx 63.66\%\] Thus, the area of the inscribed square is approximately \(63.66\%\) of the area of the circle.