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

$$L \;=\; \Delta \; (\pi \;/\; 180) \; r$$
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 Segment formula

$$A \;=\; \frac {r^2} {2} \; [\; \Delta - sin( \Delta) \;]$$

$$A \;=\; r^2 \; [\; \Delta - sin(\Delta) \;] \;/\; 2$$

$$A \;=\; r^2 \; [\; ( \Delta \; \pi \;/\; 360 ) - ( sin(\Delta) \;/\; 2 ) \;]$$

$$A \;=\; \frac{ 1 }{ 2 } \; r^2 \; [\; (\pi\;/\;180) \; \Delta - sin(\Delta) \;]$$

Symbol English Metric
$$A$$ = area $$in^2$$  $$mm^2$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$\pi$$ = Pi $$3.141 592 653 ...$$
$$r$$ = radius $$in$$ $$mm$$

Distance from Centroid of a Segment formulas

$$C_x \;=\; 0$$

$$C_y \;=\; \frac{4 \; r}{3} \; [\; sin^3 ( \frac{\Delta}{2} ) \;/\; \Delta - sin(\Delta) \;]$$

Symbol English Metric
$$C_x, C_y$$ = distance from centroid $$in$$ $$mm$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$r$$ = radius $$in$$ $$mm$$

Elastic Section Modulus of a Segment formula

$$S \;=\; I_x \;/\; C_y - r \; cos( \frac {\Delta}{2} )$$
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

$$h \;=\; r \; [\; 1 - cos ( \frac{ \Delta }{2} ) \;]$$

$$h \;=\; r - \sqrt{ r^2 - ( l^2 \;/\;4 ) }$$

Symbol English Metric
$$h$$ = height $$in$$ $$mm$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$l$$ = chord $$in$$ $$mm$$
$$r$$ = radius $$in$$ $$mm$$

Length of a Segment formulas

$$c \;=\; 2 \; r \; sin (\frac{ \Delta }{2} )$$

$$c \;=\; r \; \sqrt{ 2 - 2 \; cos( \Delta) }$$

Symbol English Metric
$$l$$ = chord $$in$$ $$mm$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$r$$ = radius $$in$$ $$mm$$

Perimeter of a Segment formula

$$P \;=\; \Delta \; r + 2\;r \; sin (\frac{\Delta }{2})$$
Symbol English Metric
$$P$$ = perimeter $$in$$ $$mm$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$r$$ = radius $$in$$ $$mm$$

Polar Moment of Inertia of a Segment formula

$$J_{z} \;=\; \frac{r^4}{4} \; [\; \Delta - sin( \Delta) + \frac{2}{3} \; sin( \Delta) \; sin^2(\frac{\Delta}{2}) \;]$$
Symbol English Metric
$$J$$ = torsional constant $$in^4$$ $$mm^4$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$r$$ = radius $$in$$ $$mm$$

Radius of Gyration of a Segment formulas

$$k_{x} \;=\; \sqrt{ I_x\;/\;A }$$

$$k_{y} \;=\; \sqrt{ I_y\;/\;A }$$

$$k_{z} \;=\; \sqrt{ k_{x}{^2} + k_{y}{^2} }$$

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

$$I_{x} \;=\; \frac {r^4}{8} \; [\; \Delta - sin( \Delta) + 2 \; sin( \Delta) \; sin^2( \frac {\Delta}{2} ) \;]$$

$$I_{y} \;=\; \frac {r^4}{24} \; [\; 3 \; \Delta - 3 \; sin( \Delta) - 2 \; sin( \Delta) \; sin^2( \frac {\Delta}{2}) \;]$$

Symbol English Metric
$$I$$ = moment of inertia $$in^4$$ $$mm^4$$
$$\Delta$$ = angle $$deg$$ $$rad$$
$$r$$ = radius $$in$$ $$mm$$