Segment of a Circle

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

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

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.

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

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.
See Article Link - Geometric Properties of Structural Shapes

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

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

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

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

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

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

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