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.

Key concepts

Circle-Square Relationship

The circle-square relationship is crucial in geometry for understanding how inscribed and circumscribed shapes interact. When a square is inscribed in a circle, all its corners (or vertices) rest on the circle's circumference. This setup means that the diagonal of the square is exactly equal to the circle's diameter. This insight helps us establish critical dimensions and relationships between the square and the circle. Understanding this foundational concept allows us to explore how the geometric attributes of both shapes influence each other and set the stage for calculating areas and understanding percentages, as the inscribed square's size directly depends on the circle's size.

Pythagorean Theorem in Geometry

The Pythagorean theorem is a fundamental principle in geometry, expressing the relationship between the sides of a right triangle. In the context of a square inscribed in a circle, the square's diagonal acts as a hypotenuse for a right triangle formed by two of the square's sides. Applying the Pythagorean theorem here allows us to calculate this diagonal:

• Let the side length be denoted as \(s\).
• The diagonal (hypotenuse) \(d\) is given by \(d^2 = s^2 + s^2\). This simplifies to \(d = \sqrt{2}s\).
• Thus, the diagonal = \(2r\), where \(r\) is the circle's radius.

This calculation directly leads us to find the side length in terms of the radius, setting the stage for further computations.

Area Calculation

Area calculation involves determining the size of a surface within a given boundary. For both the square and the circle, formulas become our tools:- **Square Area:** The area of a square is calculated as the side length squared, \(s^2\). As demonstrated, \(s = \sqrt{2}r\), leading to \(A_s = 2r^2\).- **Circle Area:** The area of a circle is given by \(\pi r^2\), where \(r\) is the radius. With both areas quantified, students can compare and relate these areas by creating ratios. Understanding these calculations helps elucidate how much space each shape covers, making it easier to appreciate differences or conversions into percentages when necessary.

Inscribed Shapes in Geometry

Inscribed shapes refer to figures placed within another figure, where all the vertices of the inscribed shape touch the boundaries of the encompassing figure. In our example, a square is inscribed in a circle. This setting offers unique geometric properties:

• The circle fully encompasses the square, providing a controlled boundary.
• The square's diagonal equals the diameter of the circle.
• All derived geometric relationships directly reflect the proportions dictated by the containing shape (in this case, the circle).

Understanding inscribed shapes not only informs us about specific size relationships but also about how to derive specific measurements and properties that are crucial for further mathematical and problem-solving applications in geometry.