Circle Basics

What is a Circle?

It is a set of all those points that are at a fixed distance from another point on the same plane. Geometry

Centre and Radius

The fixed point inside a circle that is equidistant from all the points on that circle, is called the Centre of that circle.

The line segment joining any point on the circle with its centre is called the Radius. Geometry

Chord and Diameter

The line segment joining any two points on the circle is called a Chord of that circle. Geometry

The chord that passes through the centre of a circle is the Diameter of that circle. It’s invariably the largest chord of that circle. Geometry

Length of the diameter of a circle = 2 × Length of the radius of a circle

All diameters of a circle have the same length.

Arc and Semi-circle

A piece of circle between two points is called an Arc. Geometry

The bigger arc is called the Major arc, while the smaller arc is called the Minor arc. Geometry

The arc joining the opposite points of a diameter is called the Semi-circle. There are no minor or major arcs in such a case; only two equal semi-circles. Geometry

Segment and Sectors

Segment is the region between a chord and an arc of a circle.

The segment made with major arc is called major segment. While, the segment made with minor arc is called minor segment. Geometry

Sector is the region between an arc and the two radii, joining the centre of the circle to the end points of the arc.

The sector made with major arc is called major sector. While, the sector made with minor arc is called minor sector. Geometry

When two arcs are equal, then both segments and both sectors become one and the same and each is known as a semi-circular region.

Secant and Tangent

Secant is a line that intercepts the circle at two points.

Tangent to a circle is a line that touches the circle at only one point. Geometry

The tangent to a circle is a special case of the secant.

The point where a tangent touches the circle is called the common point. Only one tangent can touch any given point on a circle.

Concentric circles

Concentric circles are two or more circles that:

are on the same plane, and
have the same centre.

Common Tangents

A tangent that touches two circles is called a common tangent.

Common tangent can be of two types:

Direct common tangent - A direct common tangent divides the line passing through centres of two circles externally in the ratio of their radii.
Transverse common tangent - A traverse common tangent divides the line passing through centres of two circles internally in the ratio of their radii.

Two circles can have :

a maximum of 4 tangents
a minimum of 0 tangents.

So, in total 5 cases are possible. Let’s see these cases.

Case 1: 4 Common tangents

Two circles, not touching each other have:

2 Direct common tangents
2 Transverse common tangents

Here, Distance between the centres of two circles > Sum of their radii (i.e. $r_1 + r_2$) Geometry

Case 2: 3 Common tangents

Two circles, touching each other externally have:

2 Direct common tangents
1 Transverse common tangent

Here, Distance between the centres of two circles = Sum of their radii (i.e. $r_1 + r_2$) Geometry

Case 3: 2 Common tangents

Two circles, intersecting each other at two points have:

2 Direct common tangents
No Transverse common tangents

Here, Difference of their radii (i.e. $|r_1 - r_2|$) < Distance between the centres of two circles < Sum of their radii (i.e. $r_1 + r_2$) Geometry

Case 4: 1 Common tangent

Two circles, touching each other internally have only 1 common tangent.

Here, Distance between the centres of two circles = Difference of their radii (i.e. $|r_1 - r_2|$) Geometry

Case 5: No Common tangents

Two circles, one inside the other but not touching have no common tangents.

Here, Distance between the centres of two circles < Difference of their radii (i.e. $|r_1 - r_2|$) Geometry

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Last updated on Sep 11, 2021