Basics of Circles

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A circle is a plane figure bounded by one curved line where all of its points are at a constant distance from a fixed point in a plane. The fixed point is called the center of the circle, and the constant distance from the center of the circle to every point on the curved line is called the radius of the circle.

Any straight line drawn through the center of the circle and ending at both ways by the circumference is called diameter. In the figure below, when the radius OA is produced to meet the circle again at B, the AB is formed.

Any two diameters intersect at the center of the circle. So, the center of the circle is the midpoint of the diameter. Therefore, length of the diameter is equal to twice the radius.

The length of the closed curve of the circle is known as the circumference of the circle. The circumference of the circle can be determined using the formula:

Circumference = 2 π r
where r is the radius of the circle. This formula can also be written as circumference = πd, where d is the diameter of the circle.

Arc of a circle
An arc is a part of circle’s circumference. An arc contains two endpoints and all of the points on the circle between the endpoints. When any two points on a circle is selected, two arcs are created: a major arc, which is the larger arc, and a minor arc, which is the shorter arc.

Chord of a circle
The line segment joining any two points on the circumference is called a chord of the circle. The chord, unless it is also a diameter, divides the circle in into two arcs. The diameter can also be defined as the chord passing through the center of the circle and is the longest chord of the circle.

Symmetrical properties
a. The perpendicular drawn from the center of a circle to a chord bisects the chord.
b. Equal chords of a circle are equidistant from the center of the circle.

Concentric circles
Circles having the same center but different radii are known as concentric circles.

Congruent circles
Two circles can be called congruent if and only if one of them can be superposed on the other to cover it completely. In other words, if the radii of two circles are of same measurement, then the two circles are congruent.

Area of a circle
The area of a circle depends on the radius of the circle. The area of a circle can be determined using the formula
Area = πr2

The portion of a circle enclosed between two radii and an arc is called a sector. If an arc subtends an angle q at the center, then the area of the sector is given by
Area of the sector = θ / 360 x πr2

Tangent of a circle
A line that intersects the circle at exactly one point is called a tangent line. The radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line.

Every point in space outside the circle can extend exactly two tangent lines to the circle. The distance from the origin of the two tangents to the points of tangency is always equal. In the following figure, XY = XZ.