## Circles

Among the several geometric shapes that we see around us in life, circle is one of the most basic and frequently seen shape.

Some of the examples of objects that we see and use which are circular in shape are the sun, the moon, a coin, caps and lids of bottles and several others.

Let us now study a circle as a geometric figure with the other necessary geometric-like concepts.

Definition of a circle

The set of all points in a plane that are a same distance from a point form a circle. A circle is a closed curve.

The set of all such points are said to form a circle. The fixed point is called centre of the circle. The same distance is called radius of the circle.

In the following figure, the point in orange is the centre of the circle, denoted by the letter O.

The closed curve in pink is the circle. It is a set of points which are all at a same distance from O, the centre of the circle.

In the following figure, a few points numbered from 1 through 8 are shown in a plane.

If a curved path is traced over all of the points, then each of them will be a same distance from the point O.

The closed curved path traversed around the points 1 through 8 is a circle. The fixed point O from which all points on the circle are at a same distance is centre of

the circle. The fixed distance is radius of the circle.

The line segment joining the centre of a circle to any point on the circle is called radius of the circle.

In the figure above, the lines, in various colors, joining point O to the numbered points are all equal in length. Each of the lines is a radius of the circle O.

CIRCUMFERENCE OF A CIRCLE

A circle is formed in a plane.

(Some examples of a plane are the computer screen, a table top, a wall, the ceiling and many other flat surfaces)

A circle divides the points in a plane into three regions.

In other words, points can lie in three different regions with respect to a circle.

In the following figure, consider the box, a rectangle (in orange) as a plane, inside which the circle in green with centre O is shown drawn.

Now, consider the three points A, B and O in the plane

Point A is exterior to the circle, i.e. outside of the circle;

Point B lies on the circle;

Point O, i.e. the centre of the circle, lies inside the circle.

Note:

• If a point is exterior to a circle, then its distance from the centre of the circle is greater than the radius of the circle;

• If a point lies on the circle, then its distance from the centre of the circle is equal to the radius of the circle;
• If a point lies inside the circle, then its distance from the centre of the circle is less than the radius of the circle.

Definitions of:

Arc, Chord, Diameter, Secant and Tangent in a circle

In the figure above, the following are displayed:

1. Arc AB
The curved path from point A to point B is the arc AB, denoted as

2. Chord AB

The line segment joining any two points on a circle is called a chord.

In the figure above, the straight line AB joining the two points A and B on the circle is a chord of the circle O.

The line segment joining point C or D on the circle and centre O of the circle is called Radius of the circle.

In the above circle, the two straight lines OC and OD are two radii of the circle.

(All radii of a circle are equal in length)

Note:

A circle can have infinitely many radii.

3. Diameter CD

The Chord passing through the centre of the circle O is called diameter of the circle.

The length of the diameter of a circle is twice the length of its radius.

If r is radius and d is diameter, then

d = 2r

Note:

1. The diameter is a longest chord in a circle.

2. A circle can have infinitely many diameters.

4. Secant

Any straight line which passes through a circle at two distinct points is called a Secant.

The straight line EF passing through the circle at two distinct points numbered 1 and 2 in the figure above is a Secant.

5. Tangent

A straight line that touches a circle exactly at one point is called a Tangent to the circle.

In the above figure, Tangent GH touches the circle at the point D.

The point at which a tangent touches a circle is called point of contact of the tangent with the circle.

In the figure, D is the point of contact of tangent GH with the circle O.

Note:

The angle between the tangent and the radius of the circle at the point of contact is 900.

We also say that the tangent to a circle and the radius of the circle are perpendicular to each other at the point of contact.