# Sector of a Circle

on . 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.
• 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.

### Angle of a Sector formula

$$\Delta \;=\; 2 \; A \;/\;r^2$$
Symbol English Metric
$$\Delta$$ = angle $$deg$$ $$rad$$
$$A$$ = area of sector $$in^2$$ $$mm^2$$
$$r$$ = radius $$in$$ $$mm$$

### 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 a Sector formula

$$r \;=\; \sqrt{ 2 \; A \;/\; \Delta }$$
Symbol English Metric
$$r$$ = radius $$in$$ $$mm$$
$$A$$ = area of sector $$in^2$$ $$mm^2$$
$$\Delta$$ = angle $$deg$$ $$rad$$

### 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 $$in$$ $$mm$$