Circle:

set of points whose distance r to center point are the same.
x and y both squared, and with same coefficient, either both positive or both negative: Ax2+Cy2+[Dx]+[Ey]-F=0, A=C
General form equation: x2+y2+Dx+Ey-F=0

Circle centered at origin (0,0):
  
Circle centered at (h,k): use distance formula: √((x-h)2+(y-k)2)=r

General form equation: x2+y2-2hx-2ky+(h2+k2-r2)=0

Area of a circle = πr2. Circumference = 2πr. d=2r, π=C/d ratio of circumference to diameter.
Double the radius → quadruple the area and double the circumference.
Circle is the shape that encloses the most area for a given "perimeter". C=1 → A=1/(4π)≈0.079577 (cf square of side 1/4 has area 1/16=.0625)