To find a function that gives the area of a square inscribed in a circle as a function of the radius of the circle, let's follow these steps:
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Understand the relationship between the square and the circle: A square inscribed in a circle has its corners touch the circumference of the circle. The center of the circle coincides with the center of the square. If we denote the radius of the circle as \( r \), we need to find the length of the side of the square in terms of \( r \).
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Determine the side length of the square: The diagonal of the square is equal to the diameter of the circle. The diameter \( D \) of the circle is given by: \[ D = 2r \] For a square with side length \( s \), the diagonal \( d \) can be expressed using the Pythagorean theorem: \[ d = s\sqrt{2} \] Since the diagonal of the square is equal to the diameter of the circle, we can set these two equations equal to each other: \[ s\sqrt{2} = 2r \] Solving for the side length \( s \): \[ s = \frac{2r}{\sqrt{2}} = r\sqrt{2} \]
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Find the area of the square: The area \( A \) of the square is given by the formula: \[ A = s^2 \] Substituting \( s = r\sqrt{2} \): \[ A = (r\sqrt{2})^2 = r^2 \cdot 2 = 2r^2 \]
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Final function: Thus, we have derived a function for the area of the square: \[ A(r) = 2r^2 \]
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Describing symmetry: The function \( A(r) = 2r^2 \) is symmetric with respect to the \( y \)-axis in a Cartesian coordinate system, as it depends only on the square of the radius \( r \). This implies that for positive and negative values of \( r \) (in the context of radius, negative does not apply, but mathematically it's still symmetrical), the area of the square will always yield non-negative results.
Summary:
The area of a square inscribed in a circle can be expressed as a function of the circle's radius \( r \) as: \[ A(r) = 2r^2 \] This function shows that the area increases with the square of the radius of the circle, demonstrating a quadratic relationship with symmetry about the vertical axis.
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