Circle is an important shape in geometry. Circles appear everywhere in the real world, from the functional gear or pulley to the edible pancake.

## Circle Definition

A circle is the set of all points in a plane that are at a fixed distance from a central point.

## Properties of a Circle

A circle is named by its center point. For example, In the given figure point C is the center of the circle The symbol of the circle is $\odot$. So this circle will be denoted as $\odot$C.

In this circle, points A, B and R are lying on the circle. Points C and Q are in the interior of the circle and points E and D are in the exterior of the circle.

The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone's symmetry axis.

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

## Center of a Circle

The central point from which all the points lying on the circle are equidistant is called as the center of the circle.

The distance from any point lying on the circle from the center is called as radius of a circle.

All radii of the circle are congruent.

In the circle P, the segment PA joining P to the point A lying on the circle , is the radius of the circle.
Letter “r” denotes radius of a circle.

## Diameter of a Circle

The distance between any two points across the circles through the center is called as diameter of the circle. Diameter equals twice the length of the radius, as we get the diameter by joining two radii from a center pointing to opposite directions.
Diameter is the maximum distance between two opposite points on the circle.

Segment AB is the diameter of the circle P.
AB can be obtained by joining two radii AP and BP end to end.
Letter “d” denotes diameter of a circle.
d = 2 r
A real life example of diameter is a medium pizza with 9 inch diameter/small pizza with 6 inch diameter.

## Circumference of a Circle

The perimeter of a circle (distance around it’s edge) is called as circumference of a circle.
If “r” is the radius and “d” is the diameter of a circle, then its circumference C is given by the formula
C = 2$\pi$r = $\pi$d

Where $\pi$ = $\frac{22}{7}$ or 3.14

## Chord of a Circle

A line segment joining two points lying on the circle is called as a chord.

Diameter is the longest chord for any circle.

## Tangent of a Circle

A line that touches the circle at one and only one point is called as tangent to the circle.

At any given point of a circle there is one and only one tangent.

In this circle only line T touches circle just once so only line T is the tangent to the circle at point P.

The tangent at any point of a circle is always perpendicular to the radius through the point of contact.

The line t is tangent to the circle at point P and perpendicular to the radius OP at P.
The lengths of tangents drawn from an external point to a circle are equal.

Ray PQ and PR make the tangents to the circle O at points Q and R respectively.
l(PQ) = l(PR).

## Arc of a Circle

An arc is a part of the circle. The intercepted arc of an angle is determined by the two points of intersection of the angle with the circle.

Measurement by central angle, The arc AB is intercepted by the central angle
$\angle$AOB. The smaller arc is called as minor arc (red arc) and the larger arc is called as major arc (blue arc)

Arcs are measured in two ways

1) As the measure of the central angle
Here, the minor arc measures 120º
The major arc measures 240º

2) The length of the arc
Measurement by arc length (radians). If r is the radius of the circle and θ is the central angle in radians, then the length of the arc can be obtained using the formula:
S = r$\theta$

Minor arc: $\theta$ = $\frac{2 \pi}{3}$, r = 2

Length of the red arc: S = $\frac{2 \pi}{3}$ $\times$ 2 = $\frac{4 \pi}{3}$

Major arc: $\theta$ = $\frac{4 \pi}{3}$, r = 2

Length of blue arc: s = 2 $\times$ 4$\frac{pi}{3}$ = 8$\frac{pi}{3}$

## Sector of a Circle

The area formed by the arc and the central angle is called as sector of a circle. The sector is enclosed by an arc and two radii. In the following diagram the yellow portion is the minor sector and unshaded part is the major sector.

Area of a sector with central angle $\theta$ and radius r is given by the formula

A = $\frac{\pi r^2 \theta}{360}$, where $\theta$ is measured in degrees.

A = $\frac{r^2}{2}$, if $\theta$ is in radians.

A circle has two special types of sectors:
Semicircle: Half of the circle.

## Segment of a Circle

The area enclosed by an arc and a chord is called as segment of a circle.

Area of the segment = AXB  = area of sector OAXB - area of $\Delta$AOB

If r is the radius and $\theta$ is the angle AOB, then area of the segment

A = $r^2$ $\frac{\theta}{2}$ – $r^2$ $\frac{sin \theta}{2}$

## Circle Problems

### Solved Examples

Question 1: Referring to the following figure name
i)    Center of the circle
ii)    The two chords that are not diameters.
iii) Minimum two radii of the circle.
iv)    A diameter of the circle

Solution:

i) Center of the circle E
ii)    The two chords that are not diameters.: AB, CD
iii) Minimum two radii of the circle. ED, ED
iv)    A diameter  of the circle: BD

Question 2: Calculate the circumference of a circle with diameter 20 cm. Take π = 3.14
Solution:

The formula for circumference is
C = 2πr = π d
Here d = 20, π = 3.14
So, C = 20 $\times$ 3.14
Circumference = 62.8 cm

Question 3: In the circle given below identify the tangent

Solution:

Segment TU is the tangent as it touches the circle at only one point.

Question 4:  Identify minor and major arc of the circle.

Solution:

Arc AXB is the minor arc as it is smaller in measure, measure of the arc AXB = 100º
Major arc = arc AYB, measure of the arc AYB = 360º - 100º = 260º