This special relationship isn't found for squares, rectangles, etc.

One reason circumference is special it that it involves, being the ratio of the diameter to the circumference.

Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary. Circumference is special to a circle (the 'circ' at the beginning is a clue), (though the phrase can be used for ellipses too).

The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be ( cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: crd ⁡ θ = ( 1 − cos ⁡ θ ) 2 + sin 2 ⁡ θ = 2 − 2 cos ⁡ θ = 2 sin ⁡ ( θ 2 ). The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). Next, substitute the values of the given point (2 for x and 11 for y), getting. So your equation starts as ( x + 1 )2 + ( y + 7 )2 r2. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The standard equation for a circle centred at (h,k) with radius r. The chord function is defined geometrically as shown in the picture. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.

In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1 / 2 to 180 degrees by increments of 1 / 2 degree. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7 + 1 / 2 degrees. Chords were used extensively in the early development of trigonometry.

Khan academy circle geometry