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   Circle Formulas   

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also see Defining Geometric Figures

  Circle

 

   
All points on the circumference of a circle
are equidistant from its center.

Radius: r
Diameter: d
Circumference: C
Area: K
 
    d = 2r
C = 2 Pi r = Pi d
 
K = Pi r2 = Pi d2/4
C = 2 sqrt(Pi K)
 
K = C2/4 Pi = Cr/2
   

 
To read about circles, visit The Geometry Center.




  Arc of a Circle

  

   
A curved portion of a circle.

Length: s
 
Central angle:
     theta (in radians),
     alpha (in degrees)

 
    s = r theta = r alpha Pi/180



  Segment of a Circle
 

 

  Either of the two regions into which a secant or a chord cuts a circle. (However, the formulas below assume that the segment is no larger than a semi-circle.)

Chord length: c
Height: h

Distance from center of circle to chord's midpoint: d
Central angle: theta (in radians), alpha (in degrees)
 
Area: K
Arc length: s
   
     theta = 2 arccos(d/r) = 2 arctan(c/(2d)) = 2 arcsin(c/(2r))
     h = r - d

     c = 2 sqrt(r2-d2) = 2r sin(theta/2) = 2d tan(theta/2) = 2 sqrt[h(2r-h)]
     d = sqrt(4r2-c2)/2 = r cos(theta/2) = c cot(theta/2)/2

     K = r2[theta-sin(theta)]/2 = r2arccos([r-h]/r) - (r-h)sqrt(2rh-h2)
         = r2arccos(d/r) - d sqrt(r2-d2)

     theta = s/r
     K = r2[s/r - sin(s/r)]/2

For much more about segments of circles, see
The Arc, Chord, Radius, Height, Angle, Apothem, and Area.



  Sector of a Circle
 

 

  The pie-shaped piece of a circle 'cut out' by two radii.

Central angle:
     theta (in radians),
     alpha (in degrees)
 
Area: K
Arc length: s

     K = r2theta/2 = r2alpha Pi/360
     theta = s/r
     K = rs/2

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