Sector of a Circle

on 27 May 2018. Posted in Plane Geometry

Sector is a fraction of the area of a circle with a radius on each side 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.
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.
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.
Segment is an interior part of a circle bound by a chord 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.

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

area of a Sector formula
\( A \;=\; \Delta \;r^2 \;/\;2 \) \( A \;=\; ( \Delta \;/\; 360 ) \; \pi \; r^2 \) \( A \;=\; ( \Delta \; \pi \;/\; 360 ) \; r^2 \)
Symbol	English	Metric
\( A \) = area of sector	\( in^2 \)	\(mm^2 \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( \pi \) = Pi	\(3.141 592 653 ...\)
\( r \) = radius	\( in \)	\( mm \)

Distance from Centroid of a Sector formulas
\( C_x \;=\; 2 \; r \; [\; sin(\Delta) \;/\;3\; \Delta \;] \) \( C_y \;=\; 0 \)
Symbol	English	Metric
\( C \) = distance from centroid	\( in \)	\( mm \)
\( A \) = area of sector	\( in \)	\( mm \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( r \) = radius	\( in \)	\( mm \)

Elastic Section Modulus of a Sector formula
\( S \;=\; I_x \;/\; sin(\Delta) \; r \)
Symbol	English	Metric
\( S \) = elastic section modulus	\( in^3 \)	\( mm^3 \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( I \) = moment of inertia	\( in^4 \)	\( mm^4 \)
\( r \) = radius	\( in \)	\( mm \)

Perimeter of a Sector formula
\( P \;=\; 2 \; r + 2 \; r \; \Delta \)
Symbol	English	Metric
\( P \) = perimeter	\( in \)	\( mm \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( r \) = radius	\( in \)	\( mm \)

Polar Moment of Inertia of a Sector formulas
\( J_{z} \;=\; (r^4\;/\;18) \; ( 9 \; \Delta^2 - 8 \; sin^2(\Delta) \;/\; \Delta ) \) \( J_{z1} \;=\; r^4 \; \Delta\;/\;2 \)
Symbol	English	Metric
\( J \) = torsional constant	\( in^4 \)	\( mm^4 \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( r \) = radius	\( in \)	\( mm \)

Radius of Gyration of a Sector formulas
\( k_{x} \;=\; \frac{1}{4} \; \sqrt{ 2 \; r^2 \; [\; 2\; \Delta - sin(2 \; \Delta ) \;/\; \Delta \;] } \) \( k_{y} \;=\; \frac{1}{12} \; \sqrt{ 2 \; r^2 \; [\; 180^2 + 9\; \Delta \; sin(2\; \Delta) - 32 + 32 \; cos^2(\Delta) \;/\; \Delta^2 \;] } \) \( k_{z} \;=\; \frac{1}{6} \; \sqrt{ 2 \; r^2 \; [\; 9 \; \Delta^2 - 8 \; sin^2(2\; \Delta ) \;/\; \Delta^2 \;] } \) \( k_{x1} \;=\; \frac{1}{4} \; \sqrt{ 2 \; r^2 \; [\; 2\; \Delta - sin(2 \; \Delta) \;/\; \Delta \;] } \) \( k_{y1} \;=\; \frac{1}{4} \; \sqrt{ 2 \; r^2 \; [\; 2\; \Delta + sin(2 \; \Delta ) \;/\; \Delta \;] } \) \( k_{x1} \;=\; r\;/\; \sqrt{2} \)
Symbol	English	Metric
\( k \) = radius of gyration	\( in \)	\( mm \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( r \) = radius	\( in \)	\( mm \)

Second Moment of Area of a Sector formulas
\( I_{x} \;=\; (r^4\;/\;4) \; [\; \Delta - \frac{1}{2} \; sin( 2 \; \Delta ) \;] \) \( I_{y} \;=\; (r^4\;/\;4) \; [\; \Delta + \frac{1}{2} \; sin( 2 \; \Delta) \;] - [\; (4\;r^4\;/\;9\; \Delta ) \; sin^2 ( \Delta ) \;] \) \( I_{x1} \;=\; I_x + r^4 \; \Delta \; sin^2 (\Delta) \) \( I_{y1} \;=\; (r^4\;/\;4) \; [\; \Delta + \frac{1}{2} \; sin( 2 \; \Delta ) \;] \)
Symbol	English	Metric
\( I \) = moment of inertia	\( in^4 \)	\( mm^4 \)
\( \Delta \) = angle	\( deg \)	\(rad \)
\( r \) = radius	\( in \)	\( mm \)

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