Proportionality Principle Between Circumferences

The lengths of circumferences are directly proportional to their radii, expressed as $$ c:c' = r:r' $$
proportionality between circumference and radius

In essence, when comparing any two circumferences, their length ratio equals the ratio of their radii.

$$ \frac{c}{c'} = \frac{r}{r'} $$

Here, c and c' denote the circumferences' lengths, while r and r' stand for the radii.

Note: This principle reveals a fundamental relationship between a circle's circumference and its radius, highlighting the significance of what we recognize today as "pi" (π=3.1415...), which is the ratio of the circumference to its diameter (twice the radius).

    Demonstrating the Principle

    Let's consider two circles with circumferences c and c', and radii r and r'.

    circumferences

    I'll illustrate this by inscribing regular polygons, such as hexagons, inside each circle.

    regular polygons inscribed in the circumferences

    Given these are regular polygons, and therefore similar polygons, their perimeters and sides maintain a consistent ratio.

    Furthermore, a theorem on regular polygons states that the radii of the circumferences they inscribe or circumscribe are proportional as well.

    This allows us to state:

    $$ 2p : 2p' = r: r' $$

    Where 2p represents the perimeter (and p the semiperimeter) of the regular polygon, and r the radius of the circle.

    The proportionality ratio, therefore, remains consistent.

    $$ k = \frac{2p}{2p'} = \frac{r}{r'} $$

    Extending this logic to regular polygons circumscribed around the circle affirms the principle.

    For example, using hexagons again but circumscribed, we find:

    regular polygons circumscribed around the circumference

    As these polygons are regular and similar, their perimeters, sides, and the circles' radii are proportionally related.

    $$ 2P : 2P' = r: r' $$

    With 2P as the perimeter (and P the semiperimeter) of the circumscribed regular polygon, and r the circle's radius.

    Knowing the perimeter of the circumscribing polygon always exceeds that of the inscribed one, we can deduce:

    $$ 2p < c < 2P $$

    Meaning, the circumference's length (c) is bounded by the perimeters of the inscribed and circumscribed polygons.

    Applying the proportionality ratio k across these relations:

    $$ 2p \cdot k < c \cdot k < 2P \cdot k $$

    Given 2p'=2p·k and 2P'=2P·k, we conclude:

    $$ 2p' < c' < 2P' $$

    Thus, a proportional relationship governs the lengths of the two circumferences, mir roring the ratio found in the inscribed and circumscribed regular polygons.

    $$ c' = c \cdot k $$

    or, equivalently,

    $$ k = \frac{c}{c'} $$

    Given k also represents the ratio between the two circles' radii.

    $$ k = \frac{c}{c'} = \frac{r}{r'} $$

    This confirms the proportion:

    $$ c : c' = r : r' $$

    Proving the direct proportionality between the circumferences' lengths and their radii.

    This logic extends to the relationship between circumferences and diameters, where the diameter is double the radius.

    $$ c : c' = 2 \cdot r : 2 \cdot r' $$

    $$ c : c' = d : d' $$

    By rearranging the means:

    $$ c : d = c' : d' $$

    In conclusion, the ratio between a circumference (c) and its diameter (d) remains constant, regardless of the circle's size, embodying the constant known as "pi" (π), valued at 3.141592...

    And that's all there is to it.

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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