Chord (c), also called long chord (LC), is between any two points on a circular curve.- Angle (\(\Delta\)) - Two rays sharing a common point.
- Back Tangent (BT) - The tangent line before the beginning of the curve.
- Center (rp) - Having all points on the line circumference are at equal distance from the center point.
- 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.
- Deflection Angle (D\(\Delta\)) - Deflection angle from full circular curve measured from tangent at PC or PT. The angles between a tangent and the ends of the chords from the PC.
- External Distance (E) - Radial distance from PI to midpoint on a simple circular curve.
- Forward Tangent (FT) - The tangent line after the ending of the curve.
- 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.
- Mid-point (M) - Center or halfway point of a line segment.
- Middle Ordinate (m) - The distance from the midpoint of the curve to the mid-point of the long chord.
- Point of Curvature (PC) - The point at which a straight line begins to curve, the point tangency to the curve
- Point of Intersection (PI) - The intersection of two tangent points or where two non-parallel lines intersect.
- Point of Tangent (PT) - The point at which a curved line ends and the point tangency to the curve begins.
- Radius (r) - Half the diameter of a circle.
- Radius Point (rp) - Radius center point of circular curve.
- 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 chord Formulas
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\( L \;=\; \Delta \cdot r \) \( L \;=\; \dfrac{ r \cdot \Delta \cdot \pi }{ 180 }\) \( L \;=\; \dfrac{ \Delta }{ 180 } \cdot 2 \cdot \pi \cdot r \) |
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| Symbol | English | Metric |
| \( L \) = length | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( \pi \) = Pi | \( 3.141 592 653 ...\) | \( 3.141 592 653 ...\) |
| \( r \) = radius | \( in \) | \( mm \) |
Length of a chord Formulas
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\( c \;=\; 2 \cdot r \cdot \sin \left( \dfrac{ \Delta}{2 } \right) \) \( c \;=\; 2 \cdot \sqrt{ r^2 - h^2 } \) |
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| Symbol | English | Metric |
| \(\large{ c }\) = chord | \( in \) | \( mm \) |
| \(\large{ h }\) = chord circle center to midpoint distance | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \( r \) = radius | \( in \) | \( mm \) |
center of circle to chord midpoint Formula
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| \( h \;=\; \sqrt{ r^2 - \dfrac{ L^2 }{ 4 } } \) | ||
| Symbol | English | Metric |
| \(\large{ h }\) = chord circle center to midpoint distance | \( in \) | \( mm \) |
| \(\large{ r }\) = circle radius | \( in \) | \( mm \) |
| \( L \) = length | \( in \) | \( mm \) |
| \(\large{ h' }\) = segment height | \( in \) | \( mm \) |
chord midpoint to arc midpoint Formula
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\( h' \;=\; r - h \) \( h' \;=\; r \cdot \left( 1 - cos \left( \dfrac{ \Delta }{ 2} \right) \;\right) \) |
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| Symbol | English | Metric |
| \(\large{ h' }\) = segment height | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \(\large{ h }\) = chord circle center to midpoint distance | \( in \) | \( mm \) |
| \(\large{ r }\) = circle radius | \( in \) | \( mm \) |
External Distance to Arc Formula
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| \( E \;=\; T \cdot tan \left( \dfrac{\Delta }{ 4 } \right) \) | ||
| Symbol | English | Metric |
| \(\large{ E }\) = external distance | \( in \) | \( mm \) |
| \( \Delta \) = angle | \( deg \) | \( rad \) |
| \(\large{ T }\) = tangent length | \( in \) | \( mm \) |
