# Segment of a Circle

on . 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.
• 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.

## 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 $$\large{ in }$$  $$\large{ mm }$$ 