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.
- See Article Links - Geometric Properties of Structural Shapes
Angle of a Sector formula |
||
| \( \Delta \;=\; \dfrac{ 2 \cdot A }{ r^2 }\) | ||
| 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 |
||
| \( L \;=\; \Delta \cdot \dfrac{ \pi }{ 180 } \cdot r \) | ||
| Symbol | English | Metric |
| \( L \) = arc length | \( in \) | \(mm \) |
| \( \Delta \) = angle | \( deg \) | \(rad \) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = radius | \( in \) | \( mm \) |
area of a Sector formula |
||
|
\( A \;=\; \dfrac{ \Delta \cdot r^2 }{ 2 }\) \( A \;=\; \dfrac{ \Delta }{ 360 } \cdot \pi \cdot r^2 \) \( A \;=\; \dfrac{ \Delta \cdot \pi }{ 360 } \cdot r^2 \) |
||
| Symbol | English | Metric |
| \( A \) = area of sector | \( in^2 \) | \(mm^2 \) |
| \( \Delta \) = angle | \( deg \) | \(rad \) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = radius | \( in \) | \( mm \) |
Distance from Centroid of a Sector formulas |
||
|
\( C_x \;=\; 2 \cdot r \cdot \dfrac{ sin(\Delta) }{ 3 \cdot \Delta } \) \( C_y \;=\; 0 \) |
||
| 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 |
||
| \( S \;=\; \dfrac{ I_x }{ sin(\Delta) \cdot r }\) | ||
| 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 |
||
| \( P \;=\; 2 \cdot r + 2 \cdot r \cdot \Delta \) | ||
| Symbol | English | Metric |
| \( P \) = perimeter | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \(rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
Polar Moment of Inertia of a Sector formulas |
||
|
\( J_{z} \;=\; \dfrac{ r^4 }{ 18 } \cdot \dfrac{ 9 \cdot \Delta^2 - 8 \cdot sin^2(\Delta) }{ \Delta }\) \( J_{z1} \;=\; \dfrac{ r^4 \cdot \Delta }{ 2 } \) |
||
| Symbol | English | Metric |
| \( J \) = torsional constant | \( in^4 \) | \( mm^4 \) |
| \( \Delta \) = angle | \( deg \) | \(rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
Radius of a Sector formula |
||
| \( r \;=\; \sqrt{ \dfrac{ 2 \cdot A }{ \Delta } } \) | ||
| 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 |
||
|
\( k_{x} \;=\; \dfrac{1}{4} \cdot \sqrt{ 2 \cdot r^2 \cdot \dfrac{ 2\cdot \Delta - sin(2 \cdot \Delta ) }{ \Delta } } \) \( k_{y} \;=\; \dfrac{1}{12} \cdot \sqrt{ 2 \cdot r^2 \cdot \dfrac{ 180^2 + 9\cdot \Delta \cdot sin(2\cdot \Delta) - 32 + 32 \cdot cos^2(\Delta) }{ \Delta^2 } } \) \( k_{z} \;=\; \dfrac{1}{6} \cdot \sqrt{ 2 \cdot r^2 \cdot \dfrac{ 9 \cdot \Delta^2 - 8 \cdot sin^2(2\cdot \Delta ) }{ \Delta^2 } } \) \( k_{x1} \;=\; \dfrac{1}{4} \cdot \sqrt{ 2 \cdot r^2 \cdot \dfrac{ 2\cdot \Delta - sin(2 \cdot \Delta) }{ \Delta } } \) \( k_{y1} \;=\; \dfrac{1}{4} \cdot \sqrt{ 2 \cdot r^2 \cdot \dfrac{ 2\cdot \Delta + sin(2 \cdot \Delta ) }{ \Delta } } \) \( k_{x1} \;=\; \dfrac{ r }{ \sqrt{2} } \) |
||
| 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 |
||
|
\( I_{x} \;=\; \dfrac{ r^4 }{ 4} \cdot \left( \Delta - \dfrac{1}{2} \cdot sin( 2 \cdot \Delta ) \right) \) \( I_{y} \;=\; \dfrac{ r^4 }{ 4} \cdot \left( \Delta + \dfrac{1}{2} \cdot sin( 2 \cdot \Delta ) \right) - \left( \dfrac{ 4 \cdot r^4 }{ 9 \cdot \Delta } \cdot sin^2 ( \Delta ) \right) \) \( I_{x1} \;=\; I_x + r^4 \cdot \Delta \cdot sin^2 (\Delta) \) \( I_{y1} \;=\; \dfrac{ r^4 }{ 4} \cdot \left( \Delta + \dfrac{1}{2} \cdot sin( 2 \cdot \Delta ) \right) \) |
||
| Symbol | English | Metric |
| \( I \) = moment of inertia | \( in^4 \) | \( mm^4 \) |
| \( \Delta \) = angle | \( deg \) | \(rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
