To determine the distance from the center to a corner of a circle given its circumference, follow these steps:

Calculate the radius of the circle: Circumference = pi x r x r, where r represents the radius of the circle. Given the circumference is known, solve for r.

Calculate the diameter of the circle: Diameter = 2r.

Divide the diameter by two to get the radius. Let's call this value "d".

Calculate the square root of d using the Pythagorean theorem: sqrt(d^2 - r^2).

Multiply the square root obtained in step 4 by 2 to obtain the length of the side of the square inscribed in the circle. Let's call this value "s".

To find the distance from the center to the corner, multiply s by the cosine of the angle between adjacent sides of the square. Since a square has four equal angles of 90 degrees, we know that the corners form a right triangle whose hypotenuse equals s. Therefore, the distance is s x cos(90 degrees) = s.

In summary, given a circle with circumference C = 3 cm, let's first calculate the radius r:
C = pi x r^2 => r = sqrt(C / pi) = sqrt(3 / pi) = 0.7682 cm
Then find the diameter D:
D = 2r => D = 2 x 0.7682 cm = 1.5364 cm
Next, divide the diameter by two to get the square root of half the diameter, d:
sqrt(D^2 - r^2) = sqrt(1.5364 cm - 0.7682 cm^2) = sqrt(0.7682 cm)
Multiplying sqrt(D^2 - r^2) by 2 gives us the length of the side of the square inscribed in the circle, s:
s = 2 x sqrt(D^2 - r^2) => s = 2 x sqrt(1.5364 cm - 0.7682 cm^2) = 1.697 cm
Finally, since the distance from the center to a corner is the product of the side length and the cosine of the angle between adjacent sides of the square, we can compute it as follows:
distance = s x cos(90 degrees) => distance = 1.697 cm x cos(90 degrees) = 1.697 cm
So the distance from the center to a corner of a circle with circumference 3 cm is approximately 1.697 cm.
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