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.

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Arc Length of a Sector 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 Sector formula
\(\large{ A = \frac{ \Delta \;r^2 }{2} }\) \(\large{ A = \frac { \Delta } { 360 } \; \pi \; r^2 }\) \(\large{ A = \frac { \Delta \; \pi } { 360 } \; r^2 }\)
Symbol	English	Metric
\(\large{ A }\) = area of sector	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ \Delta }\) = 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 Sector formulas
\(\large{ C_x = 2 \; r \; \frac{sin \; \Delta}{3\; \theta} }\) \(\large{ C_y = 0 }\)
Symbol	English	Metric
\(\large{ C }\) = distance from centroid	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ A }\) = area of sector	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

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

Perimeter of a Sector formula
\(\large{ P = 2 \; r + 2 \; r \; \Delta }\)
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 Sector formulas
\(\large{ J_{z} = \frac {r^4}{18} \; \left( \frac {9 \; \Delta^2 \;-\; 8 \; sin^2 \; \Delta }{\Delta} \right) }\) \(\large{ J_{z1} = \frac {r^4 \; \Delta}{2} }\)
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 Sector formulas
\(\large{ k_{x} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta} } }\) \(\large{ k_{y} = \frac{1}{12} \; \sqrt { 2 \; r^2 \; \frac{180^2 \; + \; 9 \; \theta \; sin \; \left(2 \; \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \theta }{\theta^2} } }\) \(\large{ k_{z} = \frac{1}{6} \; \sqrt { 2 \; r^2 \; \frac{9 \; \Delta^2 \;-\; 8 \; sin^2 \; \left(2\; \Delta \right) }{\Delta^2} } }\) \(\large{ k_{x1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \;-\; sin \; \left(2 \; \Delta \right) }{\Delta} } }\) \(\large{ k_{y1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \; + \; sin \; \left(2 \; \Delta \right) }{\Delta} } }\) \(\large{ k_{x1} = \frac{r}{ \sqrt{2} } }\)
Symbol	English	Metric
\(\large{ k }\) = radius of gyration	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \Delta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)

Second Moment of Area of a Sector formulas
\(\large{ I_{x} = \frac{r^4}{4} \; \left[ \Delta \;-\; \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] }\) \(\large{ I_{y} = \frac{r^4}{4} \; \left[ \Delta + \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] \;-\; \frac{4r^4}{9 \Delta} \; sin^2 \; \Delta }\) \(\large{ I_{x1} = I_x + r^4 \; \Delta \; sin^2 \; \Delta }\) \(\large{ I_{y1} = \frac{r^4}{4} \left[ \Delta + \frac{1}{2} \; sin \; \left( 2 \; \Delta \right) \right] }\)
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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