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The set of points whose distance to a fixed point (the center) is a fixed positive number (the radius) is a circle. A circle of radius r and center (x,y) has equation
Conversely, an equation of the form
defines a circle if d+e>f; the center is (-d, -e) and the radius is .
Three points not on the same line determine a unique circle. If the points have coordinates (x,y), (x,y) and (x,y), the equation of the circle is
A chord of a circle is a line segment between two points (Figure 1). A diameter is a chord that goes through the center, or the length of such a chord (therefore the diameter is twice the radius). Given two points P=(x,y) and P=(x,y), there is a unique circle whose diameter is PP; its equation is
The length or circumference of a circle of radius r is 2r, and the area is r. The length of the arc of circle subtended by an angle , shown as s in Figure 1, is r. Other relations between the radius, the arc length, the chord, and the areas of the corresponding sector and segment are, in the notation of (Figure 1):
Figure 1: The arc of circle subtendend by the angle is s; the chord is c; the sector is the whole slice of the pie; the segment is the cap bounded by the arc and the chord (that is, the slice minus the triangle).
Other properties of circles:
Figure 2: Left: The angle ACB equals ½ for any C in the long arc AB; and ADB equals 180°-½ for any D in the short arc AB. Right: The locus of points from which the segment AB subtends a fixed angle is an arc of circle.
(x-x)+ (y-y)+ 2(x-x)(y-y) cos =r.
A circle of radius a, passing through the pole, and with center at the point (r,)=(a,) has equation
A circle of radius a and with center at the point (r,)=(r,) has equation
Wed Oct 4 16:41:25 PDT 1995
This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.