Sector of a Circle
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.
Article Links
Angle of a Sector formula |
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\(\large{ \Delta = \frac{ 2 \; A }{r^2} }\) | ||
Symbol | English | Metric |
\(\large{ \Delta }\) = angle | \(\large{ deg }\) | \(\large{ rad }\) |
\(\large{ A }\) = area of sector | \(\large{ in^2 }\) | \(\large{ mm^2 }\) |
\(\large{ r }\) = radius | \(\large{ in }\) | \(\large{ mm }\) |
Arc Length of a Sector formula |
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\(\large{ L = \Delta \; \frac{\pi}{180} \; r }\) | ||
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 |
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\(\large{ A = \frac{ \Delta \;r^2 }{2} }\) \(\large{ A = \frac { \Delta } { 360 } \; \pi \; r^2 }\) \(\large{ A = \frac { \Delta \; \pi } { 360 } \; r^2 }\) |
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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 |
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\(\large{ C_x = 2 \; r \; \frac{sin \; \Delta}{3\; \theta} }\) \(\large{ C_y = 0 }\) |
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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 |
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\(\large{ S = \frac{ I_x }{ sin \; \Delta \; r } }\) | ||
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 |
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\(\large{ P = 2 \; r + 2 \; r \; \Delta }\) | ||
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 |
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\(\large{ J_{z} = \frac {r^4}{18} \; \left( \frac {9 \; \Delta^2 \;-\; 8 \; sin^2 \; \Delta }{\Delta} \right) }\) \(\large{ J_{z1} = \frac {r^4 \; \Delta}{2} }\) |
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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 a Sector formula |
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\(\large{ r = \sqrt{ \frac{ 2 \; A }{\Delta} } }\) | ||
Symbol | English | Metric |
\(\large{ r }\) = radius | \(\large{ in }\) | \(\large{ mm }\) |
\(\large{ A }\) = area of sector | \(\large{ in^2 }\) | \(\large{ mm^2 }\) |
\(\large{ \Delta }\) = angle | \(\large{ deg }\) | \(\large{ rad }\) |
Radius of Gyration of a Sector formulas |
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\(\large{ k_{x} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta} } }\) \(\large{ k_{y} = \frac{1}{12} \; \sqrt { 2 \; r^2 \; \frac{180^2 \; + \; 9 \; \theta \; sin \; \left(2 \; \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \theta }{\theta^2} } }\) \(\large{ k_{z} = \frac{1}{6} \; \sqrt { 2 \; r^2 \; \frac{9 \; \Delta^2 \;-\; 8 \; sin^2 \; \left(2\; \Delta \right) }{\Delta^2} } }\) \(\large{ k_{x1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \;-\; sin \; \left(2 \; \Delta \right) }{\Delta} } }\) \(\large{ k_{y1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \Delta \; + \; sin \; \left(2 \; \Delta \right) }{\Delta} } }\) \(\large{ k_{x1} = \frac{r}{ \sqrt{2} } }\) |
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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 |
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\(\large{ I_{x} = \frac{r^4}{4} \; \left[ \Delta \;-\; \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] }\) \(\large{ I_{y} = \frac{r^4}{4} \; \left[ \Delta + \frac{1}{2} \; sin \left( 2 \; \Delta \right) \right] \;-\; \frac{4r^4}{9 \Delta} \; sin^2 \; \Delta }\) \(\large{ I_{x1} = I_x + r^4 \; \Delta \; sin^2 \; \Delta }\) \(\large{ I_{y1} = \frac{r^4}{4} \left[ \Delta + \frac{1}{2} \; sin \; \left( 2 \; \Delta \right) \right] }\) |
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Symbol | English | Metric |
\(\large{ I }\) = moment of inertia | \(\large{ in^4 }\) | \(\large{ mm^4 }\) |
\(\large{ \Delta }\) = angle | \(\large{ deg }\) | \(\large{ rad }\) |
\(\large{ r }\) = radius | \(\large{ in }\) | \(\large{ mm }\) |
Tags: Inertia Equations Structural Steel Equations Modulus Equations