LESSON 6:Circles

Circle is one favorite plane geometry problem topic. Unlike polygons, all circles are of the same shape and they vary only in sizes. Circles have certain basic characteristics which will be studied in this section.

The line joining the center of a circle to any points on the circle is known as the radius. The radius is perhaps the most important measurement of a circle because once it is known, all other characteristics of the circle such as circumference and area can be determined.

The interior of the circle is the set of all points within the boundary of the circle whose distances from the center are always less than the measure of the radius.

The exterior of the circle is the set of all points outside the boundary of the circle whose distances from the center are always greater than the radius. Thus, we say that if a point P is interior to the circle then P is in the circle. If P is an exterior point then P is outside the circle. If P is neither interior nor exterior to the circle, then P is on the circle whose distance from the center is equal to the radius.

An arc is a portion of a circle that contains two endpoints and all the points on the circle between the endpoints. By choosing any two points on the circle, arcs will be formed; a major arc (the longer arc), and a minor arc (the shorter one).

A chord is a line segment joining any two points on the circle. The chord that passes through the center of the circle is called the diameter of a circle. The diameter, which is twice the length of the radius is also known as the longest chord of the circle. A chord divides the circle into two regions, the major segment and the minor segment.

A sector is the figure formed by two radii and an included arc. The central angle is the angle is the angle whose vertex lies at the center of the circle and whose sides are the two radii. The inscribed angle is the angle whose vertex lies on the circle and whose two sides are chords of the circle.

If circles of different radii have common center then they are referred to as concentric circles. The region bounded by any two concentric circles is known as the annulus. The shaded region in the figure at the left is an annulus region.

Definitions Related to Circles

arc: a curved line that is part of the circumference of a circle chord: a line segment within a circle that touches 2 points on the circle. circumference: the distance around the circle. diameter: the longest distance from one end of a circle to the other. origin: the center of the circle pi (): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle. radius: distance from center of circle to any point on it. sector: is like a slice of pie (a circle wedge). tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.

Diameter = 2 x radius of circle

Circumference of Circle = PI x diameter = 2 PI x radius
    where PI =  = 3.141592...

Area of Circle:
    area = PI r² 

Length of a Circular Arc: (with central angle )
    if the angle  is in degrees, then length =  x (PI/180) x r
    if the angle  is in radians, then length = r x 

Area of Circle Sector: (with central angle )
    if the angle  is in degrees, then area = (/360)x PI r²
    if the angle  is in radians, then area = ((/(2PI))x PI r²

Equation of Circle: (Cartesian coordinates)
  for a circle with center (j, k) and radius (r):
    (x-j)^{^2} + (y-k)^{^2} = r^{^2}

Equation of Circle: (polar coordinates)
    for a circle with center (0, 0):   r() = radius

    for a circle with center with polar coordinates: (c, ) and radius a:
      r² - 2cr cos( - ) + c² = a²

Equation of a Circle: (parametric coordinates)
    for a circle with origin (j, k) and radius r:
      x(t) = r cos(t) + j       y(t) = r sin(t) + k