Segment Calculator
“Segment” can mean different things. This page gives you three powerful calculators in one: circle segment (geometry), woodturning segments (shop), and SMS message segments (telecom).
Find area, arc length, chord, and central angle of a circular segment.
same unit everywhere
measured from chord
Segment formulas used
Circular segment (r & h): Central angle \( \theta = 2\arccos\left(\frac{r-h}{r}\right) \), area \( A = \frac{r^2}{2}(\theta - \sin\theta) \), chord \( c = 2\sqrt{2rh - h^2} \), arc \( L = r\theta \), perimeter \( P = L + c \).
Circular segment (r & chord c): \( \theta = 2\arcsin\left(\frac{c}{2r}\right) \), \( h = r - \sqrt{r^2 - (c/2)^2} \).
Woodturning segments: outer circumference \( C_o = \pi \cdot D_o \); segment outer length \( s_o = C_o / N \); miter angle = \( 180^\circ / N \); inner diameter \( D_i = D_o - 2 \cdot w \).
SMS segments (simplified): GSM: 160 chars → 1 segment, >160 → ceil(len/153); Unicode: 70 chars → 1 segment, >70 → ceil(len/67).
FAQs
Which tab should I use?
If you’re doing geometry (like circle cutouts or landscaping), use “Circle Segment”. If you’re building turned bowls or segmented rings, use “Woodturning”. If you’re sending SMS campaigns, use “SMS Segment”.
Why do I get NaN for circle segment?
Check that the height is less than the radius, and that the chord is not longer than the diameter. The formulas assume a valid circular segment.
Formula (LaTeX) + variables + units
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Circular segment (r & h): Central angle \( \theta = 2\arccos\left(\frac{r-h}{r}\right) \), area \( A = \frac{r^2}{2}(\theta - \sin\theta) \), chord \( c = 2\sqrt{2rh - h^2} \), arc \( L = r\theta \), perimeter \( P = L + c \). Circular segment (r & chord c): \( \theta = 2\arcsin\left(\frac{c}{2r}\right) \), \( h = r - \sqrt{r^2 - (c/2)^2} \). Woodturning segments: outer circumference \( C_o = \pi \cdot D_o \); segment outer length \( s_o = C_o / N \); miter angle = \( 180^\circ / N \); inner diameter \( D_i = D_o - 2 \cdot w \). SMS segments (simplified): GSM: 160 chars → 1 segment, >160 → ceil(len/153); Unicode: 70 chars → 1 segment, >70 → ceil(len/67).
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.