Quantitative Aptitude > Circles

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  • Given a point O in a plane and a positive number r, the set of points in the plane that are a distance of r units from O is called a circle. The point O is called the center of the circle and the distance r is called the radius of the circle. The diameter of the circle is twice the radius. Two circles with equal radii are called congruent circles.
  • Any line segment joining two points on the circle is called a chord. The terms "radius" and "diameter" can also refer to line segments: A radius is any line segment joining a point on the circle and the center of the circle, and a diameter is a chord that passes through the center of the circle.

In the figure below, O is the center of the circle, r is the radius, PQ is a chord, and ST is a diameter.

  • The distance around a circle is called the circumference of the circle, which is analogous to the perimeter of a polygon. The ratio of the circumference C to the diameter d is the same for all circles and is denoted by the Greek letter ; that is,

The value of  is approximately 3.14 and can also be approximated by the fraction  .

If r is the radius of a circle, then , and so the circumference is related to the radius as follows.

For example, if a circle has a radius of 5.2, then its circumference is

which is approximately 32.7.

  • Given any two points on a circle, an arc is the part of the circle containing the two points and all the points between them. Two points on a circle are always the endpoints of two arcs. It is customary to identify an arc by three points to avoid ambiguity. In the figure below, arc ABC is the shorter arc between A and C, and arc ADC is the longer arc between A and C.

  • A central angle of a circle is an angle with its vertex at the center of the circle. The measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc. An entire circle is considered to be an arc with measure . In the figure above, the measure of arc ABC is  and the measure of arc ADC is .
  • To find the length of an arc of a circle, note that the ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to . The circumference of the circle above is . Therefore,

=  

  • The area of a circle with radius  is equal to . For e.g. the area of the circle above with radius  is .
  • A sector of a circle is a region bounded by an arc of the circle and two radii. In the circle above, the region bounded by arc ABC and the two dashed radii is a sector with central angle . Just as in the case of the length of an arc, the ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to . Therefore, if S represents the area of the sector with central angle , then

  • A tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency, denoted by P in the figure at the top of the following page. If a line is tangent to a circle, then a radius drawn to the point of tangency is perpendicular to the tangent line. The converse is also true; that is, if a line is perpendicular to a radius at its endpoint on the circle, then the line is a tangent to the circle at that endpoint.

  • A polygon is inscribed in a circle if all its vertices lie on the circle, or equivalently, the circle is circumscribed about the polygon. Triangles RST and XYZ below are inscribed in the circles with centers O and W, respectively.

  • If one side of an inscribed triangle is a diameter of the circle, as in triangle XYZ above, then the triangle is a right triangle. Conversely, if an inscribed triangle is a right triangle, then one of its sides is a diameter of the circle.
  • A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon. In the figure below, quadrilateral ABCD is circumscribed about the circle with center O.

  • Two or more circles with the same center are called concentric circles, as shown in the figure below.

 

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