Chord Length Circumference Circle Segments

Overview

Engineering reference data for Chord Length Circumference Circle Segments in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Practical Application Example

For a circle with radius $r = 3 \, \text{m}$ divided into $n = 24$ equal segments:

1. Retrieve the unit chord length: From the table above, for $n = 24$, the chord length for a unit circle is $L_{\text{unit}} = 0.2611$.

2. Scale to actual radius: The actual chord length is: $L = L_{\text{unit}} \times r = 0.2611 \times 3 \, \text{m} = 0.7833 \, \text{m}$

3. Calculate total chord length: The total length of all 24 chords is: $L_{\text{total}} = n \times L = 24 \times 0.7833 \, \text{m} = 18.7959 \, \text{m}$

4. Compare to circumference: The exact circumference is $C = 2\pi r = 2\pi \times 3 \, \text{m} \approx 18.8496 \, \text{m}$. As the number of segments increases, the total chord length approaches the circumference.

Key Relationship

The chord length $L$ for a circle of radius $r$ with $n$ equal segments is given by the formula: $L = 2r \sin\left(\frac{\pi}{n}\right)$ where $\theta = \frac{2\pi}{n}$ is the central angle in radians. The table provides the factor $2 \sin(\pi/n)$ for a unit circle.

Interactive Charts

References

• Original Source