Chord of a Circle

Written by Jerry Ratzlaff on . Posted in Geometry

•  A line segment on the interior of a circle.
• An angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.

Chord Angle of a Circle Formula

$$\large{ m\theta_{1} = \frac{1}{2} \; \left( m\overset{\frown}{AC} + m\overset{\frown}{EG} \right) }$$

$$\large{ m\theta_{2} = \frac{1}{2} \; \left( m\overset{\frown}{CE} + m\overset{\frown}{GA} \right) }$$

$$\large{ m\theta_{1} = m\theta_{3} }$$

$$\large{ m\theta_{2} = m\theta_{4} }$$

Where:

$$\large{ \theta }$$ = angle

$$\large{ \frown }$$ = chord arc length

Chord Arc Length of a Circle Formula

$$\large{ l = \frac { \theta} { 180 } \; 2 \; \pi \; r }$$

Where:

$$\large{ l }$$ = length

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle

$$\large{ \pi }$$ = Pi

Chord Length of a Circle formula

$$\large{ c = 2 \; r \; \sin \; \frac {\theta}{2} }$$

$$\large{ c = 2 \; \sqrt{r^2-h^2} }$$

Where:

$$\large{ c }$$ = chord

$$\large{ h, h' }$$ = height

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle

Chord Tangent of a Circle Formula

$$\large{ m\theta_{1} = \frac{1}{2} \; \left( m\overset{\frown}{ABC} \right) }$$

$$\large{ m\theta_{2} = \frac{1}{2} \; \left( m\overset{\frown}{EAZ} \right) }$$

$$\large{ m\theta_{3} = \frac{1}{2} \; \left( m\overset{\frown}{GAB} \right) }$$

Where:

$$\large{ \theta }$$ = angle

$$\large{ \frown }$$ = chord arc length

chord circle center to midpoint distance Formula

$$\large{ h_c = r - h_s }$$

$$\large{ h_c = \sqrt{ r^2 - \frac{ c^2 }{ 4 } } }$$

Where:

$$\large{ h_c }$$ = chord circle center to midpoint distance

$$\large{ l_c }$$ = chord length

$$\large{ r }$$ = circle radius

$$\large{ h_s }$$ = segment height