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Circle                                                              All points on the circumference of a circle                                                       are equidistant from its center.                                                          Radius: r                                                          Diameter: d 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                                                        Circumference: C                                                       Area: K

                            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

  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