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.
- Radius Point (rp) - Radius center point of circular curve.
- 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.
- See Article Link - Geometric Properties of Structural Shapes
Arc Length of a Segment formula |
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| \( L \;=\; \Delta \cdot \dfrac{ \pi }{ 180 } \cdot r \) | ||
| 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 Segment formula |
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\( A \;=\; \dfrac{ r^2 }{2 } \cdot (\; \Delta - sin( \Delta) \;) \) \( A \;=\; \dfrac{ r^2 \cdot (\; \Delta - sin (\Delta) \; ) }{ 2 } \) \( A \;=\; r^2 \cdot \left( \dfrac{ \Delta \cdot \pi }{ 360 } - \dfrac{ sin(\Delta) }{ 2 } \right) \) \( A \;=\; \dfrac{ 1 }{ 2 } \cdot r^2 \cdot \left( \dfrac{ \pi }{ 180 } \cdot \Delta - sin(\Delta) \right) \) |
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| Symbol | English | Metric |
| \( A \) = area | \( in^2 \) | \( mm^2 \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( \pi \) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( r \) = radius | \( in \) | \( mm \) |
Distance from Centroid of a Segment formulas |
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\( C_x \;=\; 0 \) \( C_y \;=\; \dfrac{ 4 \cdot r }{ 3 } \cdot \dfrac{ sin^3 \dfrac{ \Delta }{ 2 } }{ \Delta - sin(\Delta) } \) |
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| 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 |
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| \( S \;=\; \dfrac{ I_x }{ C_y - r \cdot cos \left( \dfrac{ \Delta }{ 2 } \right) } \) | ||
| 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 |
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\( h \;=\; r \cdot \left( 1 - cos \left(\dfrac{ \Delta }{ 2 } \right) \right) \) \( h \;=\; r - \sqrt{ r^2 - \dfrac{ l^2 }{ 4 } } \) |
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| Symbol | English | Metric |
| \( h \) = height | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( l \) = chord | \( in \) | \( mm \) |
| \( r \) = radius | \( in \) | \( mm \) |
Length of a Segment formulas |
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\( c \;=\; 2 \cdot r \cdot sin \left( \dfrac{ \Delta }{ 2 } \right) \) \( c \;=\; r \cdot \sqrt{ 2 - 2 \cdot cos ( \Delta ) } \) |
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| Symbol | English | Metric |
| \( l \) = chord | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
Perimeter of a Segment formula |
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| \( P \;=\; \Delta \cdot r + 2 \cdot r \cdot sin \left( \dfrac{ \Delta }{ 2 } \right) \) | ||
| Symbol | English | Metric |
| \( P \) = perimeter | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
Polar Moment of Inertia of a Segment formula |
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| \( J_{z} \;=\; \dfrac{r^4}{4} \cdot \left( \Delta - sin( \Delta) + \dfrac{2}{3} \cdot sin( \Delta) \cdot sin^2 \left( \dfrac{\Delta}{2} \right) \right) \) | ||
| 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 |
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\( k_{x} \;=\; \sqrt{ \dfrac{ I_x }{ A } } \) \( k_{y} \;=\; \sqrt{ \dfrac{ I_y }{ A } } \) \( k_{z} \;=\; \sqrt{ k_{x}{^2} + k_{y}{^2} } \) |
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| 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 |
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\( I_{x} \;=\; \dfrac{ r^4 }{ 8 } \cdot \left( \Delta - sin( \Delta) + 2 \cdot sin( \Delta) \cdot sin^2 \left( \dfrac{ \Delta }{ 2 } \right) \right) \) \( I_{y} \;=\; \dfrac{ r^4 }{ 24 } \cdot \left( 3 \cdot \Delta - 3 \cdot sin( \Delta) - 2 \cdot sin( \Delta) \cdot sin^2 \left(\dfrac{ \Delta }{ 2 } \right) \right) \) |
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| Symbol | English | Metric |
| \( I \) = moment of inertia | \( in^4 \) | \( mm^4 \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
