Segment of a Circle

on 17 April 2018. Posted in Plane Geometry

Segment is an interior part of a circle bound by a chord and an arc.
Angle (\(\Delta\)) - Two rays sharing a common point.
Center (cp) - Having all points on the line circumference are at equal distance from the center point.
Chord (c) - Also called long chord (LC), is between any two points on a circular curve.
Circle - All points are at a fixed equal distance from a radius point (rp).
Circumference (C) - The outside of a circle or a complete circular arc.
Height (h) - Length of radius from radius center to midpoint of chord.
Height (h') - Length of radius from midpoint of chord to point on circular curve.
Length (L) - Total length of any circular curve measured along the arc.
Major Arc - The longest of two arcs of a circle or ellipse.
Minor Arc - The shorter of two arcs of a circle or ellipse.
Radius (r) - Half the diameter of a circle. A line segment between the center point and a point on a circle or sphere.
Radius Point (rp) - Radius center point of circular curve.
Sector is a fraction of the area of a circle with a radius on each side and an arc.
Tangent (T) - A line that touches a curve at just one point such that it is perpendicular to a radius line of the curve.

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Arc Length of a Segment formula
\(\large{ L = \Delta \; \frac{\pi}{180} \; r }\)
Symbol	English	Metric
\(\large{ L }\) = arc length	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Area of a Segment formula
\(\large{ A = \frac {r^2} {2} \; \left( \Delta - sin \; \Delta \right) }\) \(\large{ A = \frac {r^2 \; \left( \Delta \;-\; sin \; \Delta \right) }{ 2 } }\) \(\large{ A = r^2 \; \left( \frac { \Delta \; \pi }{ 360 } \;-\; \frac { sin \; \Delta }{ 2 } \right) }\) \(\large{ A = \frac { 1 }{ 2 } \; r^2 \; \left( \; \frac {\pi}{180} \Delta \;-\; sin \; \Delta \; \right) \;\; }\)
Symbol	English	Metric
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Distance from Centroid of a Segment formulas
\(\large{ C_x = 0 }\) \(\large{ C_y = \frac {4 \; r}{3} \; \left( \frac {sin^3 \; \frac{\Delta}{2} } {\Delta \; - \; sin \; \Delta} \right) }\)
Symbol	English	Metric
\(\large{ C_x, C_y }\) = distance from centroid	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Elastic Section Modulus of a Segment formula
\(\large{ S = \frac{ I_x }{ C_y \;-\; r \; cos \; \left( \frac {\Delta}{2} \right) } }\)
Symbol	English	Metric
\(\large{ S }\) = elastic section modulus	\(\large{ in^3 }\)	\(\large{ mm^3 }\)
\(\large{ C_x, C_y }\) = distance from centroid	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ I }\) = moment of inertia	\(\large{\frac{lbm}{ft^2-sec} }\)	\(\large{\frac{kg}{m^2} }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Height of a Segment formulas
\(\large{ h = r \; \left( 1 - cos \; \frac{ \Delta }{2} \right) }\) \(\large{ h = r - \sqrt{ r^2 - \frac{ l^2 }{4} } }\)
Symbol	English	Metric
\(\large{ h }\) = height	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ l }\) = chord	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Length of a Segment formulas
\(\large{ c = 2 \; r \; sin \; \frac{ \Delta }{2} }\) \(\large{ c = r \; \sqrt{ 2 - 2 \; cos \; \Delta } }\)
Symbol	English	Metric
\(\large{ l }\) = chord	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Perimeter of a Segment formula
\(\large{ P = \Delta \; r + 2r \; sin \; \frac { \Delta }{2} }\)
Symbol	English	Metric
\(\large{ P }\) = perimeter	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Polar Moment of Inertia of a Segment formula
\(\large{ J_{z} = \frac {r^4}{4} \; \left( \Delta - sin \; \Delta + \frac {2}{3} \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2} \right) }\)
Symbol	English	Metric
\(\large{ J }\) = torsional constant	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Radius of Gyration of a Segment formulas
\(\large{ k_{x} = \sqrt { \frac {I_x}{A} } }\) \(\large{ k_{y} = \sqrt { \frac {I_y}{A} } }\) \(\large{ k_{z} = \sqrt { k_{x}{^2} + k_{y}{^2} } }\)
Symbol	English	Metric
\(\large{ k }\) = radius of gyration	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ A }\) = area	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ I }\) = moment of inertia	\(\large{ in^4 }\)	\(\large{ mm^4 }\)

Second Moment of Area of a Segment formulas
\(\large{ I_{x} = \frac {r^4}{8} \; \left( \Delta - sin \; \Delta + 2 \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2} \right) }\) \(\large{ I_{y} = \frac {r^4}{24} \; \left( 3 \; \Delta - 3 \; sin \; \Delta - 2 \; sin \; \Delta \; sin^2 \; \frac {\Delta}{2} \right) }\)
Symbol	English	Metric
\(\large{ I }\) = moment of inertia	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

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