Question

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A circle is inscribed in a square. Write and simplify an expression for the ratio of the area of the square to the area of the circle. For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square.

Answer 1

Let's let the side length of the square be "s". Therefore, the diameter of the circle is also "s".

The area of the square is s^2.

The area of the circle is πr^2, where r is the radius of the circle. Since the diameter is equal to the side length of the square, the radius is half of the diameter, which is s/2.

So, the area of the circle is π(s/2)^2 = π(s^2/4) = (π/4)s^2.

Therefore, the ratio of the area of the square to the area of the circle is:
s^2 / (π/4)s^2
= 4 / π

So, the simplified ratio is 4/π.