__Circles:__

A circle is a plane figure, which is formed by the set of all those points which are equidistant from a fixed point in same plane. The fixed point is called the **center** of the circle and the fixed distance from the center is called the radius of the circle.

Hence, a circle can be defined as locus of a point moving in a plane, in such a manner that its distance from a fixed point is always constant.

In fig. 1, ‘*r*‘ is radius of circle and ‘*O*‘ is its center.

In fig. 2, several points are marked which lie either outside the circle or inside the circle or on the circle. Based on this any point can be defined as:

**Exterior Point:**Set of all those points in the plane of the circle whose distance from its center is greater than radius of the circle are exterior points.

A point X is exterior point w.r.t to circle with center ‘O’ if **OX** > r

In fig. 2 A, D, G, B are exterior points.

**Interior Points:**Set of all those points in the plane of the circle whose distance from its center is less than radius of the circle are interior points.

A point X is interior point w.r.t to circle with center ‘O’ if **OX** < r

In fig. 2 C, F and E are interior points.

**Point on circumference of circle:**Set of all those points whose distance from center of circle is equal to radius of the circle. In simple words, set of points lying on the circle are points on circumference of circle.

A point X is said to lie on circumference of circle with center ‘O’ if **OX = r**

In fig. 2, points P,S and R lie on circumference and on joining these points with center, i.e. OR, OP and OS 0 will represent the radius of the given circle.

Circumference of a circle is given as: C = 2*πr* units, where r is radius of circle.

Area of a circle:A = *π*r^{2} sq. units.

Now that we had a discussion about the area of the circle, let us look at a different situation. What happens if a circle and a line are considered? According to the relative positions of the line and the circle, three possibilities arise as shown in the figure given below:

In fig. 3.1, \(\overleftrightarrow{AB}\) intersects the circle at two distinct points P and Q. In such a case line \(\overleftrightarrow{AB}\) is called *secant* of the circle. Points P and Q lie on circumference of circle, but it does not pass through center ‘O’, hence line segment PQ is known as a chord of the circle as its endpoints lie on the circle.

Chord can be seen as a line segment joining any two distinct points on circle’s circumference. A chord passing through center of circle is known as diameter of the circle and it is the largest chord of the circle. This diameter is twice that of the radius i.e. D = 2r, where D is the diameter and r is the radius.

In fig. 3.2, \(\overleftrightarrow{AB}\) touches the circle exactly at one point, P. Such a line is called the *tangent* to the circle.

In fig. 3.3, \(\overleftrightarrow{AB}\) does not touch the circle at any point, therefore it is called as a non-intersecting line.

The relationship between a line and a circle must be clear by now. Now let us discuss about the circular region which is cut off from rest of the circle by a secant or a chord.** **

**Arc of circle:**

A part of circumference of circle is known as an arc.It is a continuous piece of the circle.

The arc PAQ is known as minor arc and arc PBQ is major arc.

Part of circle bounded by a chord and an arc is known as segment of circle.

Sector of a circle is the part bounded by two radii and an arc of circle.In fig. 6 AOB is a sector of circle with O as center.

The fig. 7 given below illustrates the various terms related to circles as explained above.