Circle

Geometry circles are the collection of all the points in a plane, which are at a fixed distance from the fixed point in the plane, is called a circle and is one geometry construction which belongs to the category of geometry basics. Circle

The fixed point is called the center of the circle and the fixed distance is called the radius of the circle. In the figure, O is the center and the length OP is the radius of the circle.

A geometry circle divides the plane on which it lies into three parts. They are

(i) Inside the circle, which is also called the interior of the circle, which is also called interior of the circle;

(ii) Outside the circle, which is also called the exterior of the exterior of the circle. The circle and its interior make up the circular region.

(iii) Outside the circle, which is also called the exterior of the circle. The circle and its interior make up the circular region.

(iv) The circle and its up the circular region.

There are different kinds of circles, they are:

1. Tangent Circle
2. Intersecting Circle

Tangent Circles

Tangent circles are circles to which a straight line is drawn such at it touches the circumference of the circle at one single point. The straight line is called as tangent and hence this kind of circle is called as tangent circle.

Circle with tangent is as shown in the below diagram.

Tangent Circle

Intersecting Circles

Two or more circles intersecting each other at minimum point points is called as intersecting circles. Intersecting circles are used to represent venn diagrams under the topic of sets and denoting union and intersection of sets.

Below diagram represent the intersecting circles which are used in solving geometry homework help

Intersecting Circles

Unit of a Circle

A locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle.
The fixed point is the centre and the fixed distance is called the radius of the circle.
Unit of a Circle
A unit circle is a circle with radius of 1 unit.

Area of Circle

The area of a circle is the area inside the circle enclosed by its circumference.
Circumference is the perimeter of a circle
Area of a circle = $\pi$ r²($\pi$=pi taken as $\frac{22}{7}$ or 3.142, r=radius of the circle)

Circumference of a Circle

The perimeter or length of boundary of a circle is called its Circumference.
Circumference of a circle = 2 $\pi$r
($\pi$ is taken as 3.142 approx, r = radius of the circle)

Unit Circle

A locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle.
The fixed point is the centre and the fixed distance is called the radius of the circle.A unit circle is a circle with radius of 1 unit.

Unit Circle

Diameter of a Circle

Diameter is the chord in circle passing through the center of the circle. It is denoted by the letter ‘d’. It is the longest chord in a circle.
Diameter = d = 2 x radiusDiameter of a Circle
C is the centre of the circle

Diameter is the chord in circle passing through the centre of the circle. It is denoted by the letter ‘d’. It is the longest chord in a circle.

Diameter = d = 2 x radius

(diameter is twice the radius of a circle)

Circumference of Circle

The perimeter of a circle is the circumference of the circle.
Circumference of a circle = 2 $\pi$ r($\pi$ is taken as 3.142 approx, r = radius of the circle) 

Circle Formulas

The different formula for the Circumference and area of a circle.

For a circle of radius r, we have:
    1. Circumference of the circle = 2 $\pi$ r
     2. Area of the circle = $\pi$  r2
     3. Area of the semicircle = $\frac{1}{2}$ $\pi$  r2 
     4. Perimeter of the semicircle = ($\pi$  r + 2 r)

Semi Circle

A diameter in a circle divides a circle into two equal arcs. Each of these two arcs is called a semi circle. In simple words a semi-circle is a half circle.
Semi Circle
Perimeter of a semi-circle is half the circumference of a circle.
Circumference of a semi-circle =  ($\pi$ r+ 2r)
($\pi$ =pi taken as 3.142 approx., r=radius of the circle)
Area of a semi circle is half the area of the given circle.
Area of a semi circle = $\frac{(Area of the circle)}{2}$
                            = $\frac{(\pi r^{2})}{2}$  

Perimeter of a Circle

The perimeter or length of boundary of a circle is called its Circumference.
Circumference of a circle = 2 $\pi$ r($\pi$  is taken as 3.142 approx, r = radius of the circle)

Chord of a Circle

Chord of a Circle
A line segment joining any two points on a circle is called a chord of the circle. In the above figure, AB, ST and PCQ are the three chords of a circle with centre C. A chord of a circle passing through its centre is called a diameter of the circle. Diameter is the longest chord of a circle. In the above figure, PCQ is a diameter.

Tangent of a Circle

A line meeting a circle only and only in one point is called a tangent to the circle at that point.
The point of which the tangent line meets the circle is called the point of contact.
Tangent of a Circle
In the above figure, c is the centre of the circle, r is the radius and P is the point of contact of the tangent.
The radius of the circle is always perpendicular to the tangent of that circle

Number of tangents to a circle:
     1.  There is no tangent passing through a point lying inside the circle.
     2.  There is one and only one tangent passing through a point lying on a circle
     3.  There are exactly two tangents through a point lying outside a circle

Length of tangent:
The length of the line segment of the tangent between a given point of contact with the circle is called the length of the tangent from the point to the circle.

Radius of a Circle

A line segment joining the centre of a circle and a point on the circle is called a radius of the circle.
Radius of a Circle
In the above figure, C is the centre of the circle.' r' is the radius from the centre to the different points A,B and D on the circumference
Plural of radius is called a radii. In the above figure CA,CB and CD are three radii of the circle.

Arc Length

Arc A continuous piece of a circle is called an arc of the circle.
Arc Length
In the figure above, C is the centre of the circle $\stackrel\frown{AB}$ is the arc. This arc consists of two end points A and B and all the points between these two points.
An arc is denoted by the symbol $'\stackrel\frown{}'$ above the points denoting the arc.

For example, arc AB is denoted as $\stackrel\frown{AB}$
There are three types of arcs, minor arc, major arc and a semi circle (half circle)

Semi Circle: A diameter in a circle divides a circle into two equal arcs. Each of these two arcs is called a semi circle. In simple words a semi-circle is a half circle.
Semicircle Diameter
Minor Arc: An arc which is less than a semicircle is called the minor arc. It is denoted by only the two end points. In the figure below,$\stackrel\frown{AB}$ is an arc less than the semicircle and hence it is the minor arc.
Major Arc: An arc which is more than a semicircle is called the major arc. It is denoted by the two end points and a point in between these two points on the arc. In the figure given below $\stackrel\frown{APB}$ is the major arc. Points A and B, the first and the third points are the endpoints and the middle point P on the arc is the third point. And the arc is read as major Arc APB.
Central Angle
Central angle: An angle subtended by an arc at the centre of a circle is called its central angle
Arc length: The length of the arc is the length of the part of the circle calculated using the radius  of the circle and the central angle subtended by the arc.

Formula for Arc length of a circle when central angle given is in radians
 s = r x θ¸
(s=arc length, r=radius of the circle, θ=central angle in radians)

 Formula for arc length when central angle is in degrees
s = Circumferenc x $\frac{0}{360^{0}}$
Arc length,   s = 2 $\pi$ r ($\frac{0}{360^{0}}$)

Sector of a Circle


Sector of a Circle
Sector of a circle: A sector of a circle is the part of the circle enclosed by the two radii and the intercepted arc. It looks like the shape of a circular pizza piece.
Major Sector of a Circle
If the arc is the minor arc then the sector formed is the minor sector. The remaining part of the circle is called the major sector of the circle.
The area inside this part is the area of the sector.

The angle at the centre of a circle is 360°, in other words 2$\pi$ ($\pi$ =180°)
Area of the circle is $\pi$r², so the area of a sector with some angle θ less than 2$\pi$  can be written as:
$(\frac{0}{2\pi })$x  $\pi$ r²On further simplification, we get, ($\frac{0}{2}$) x r²
Area of a sector = ($\frac{1}{2}$) x θ x r²
                          (when θ given in radians)
Area of a sector = ($\frac{1}{2}$) x (θ x $\frac{\pi }{180}$) x r²
                          (when θ given in degrees)

Segment of a Circle

Segment of a circle: The part of the circular region enclosed by an arc and a chord, this region including the arc and the chord is the segment of the circle.
Segment of a Circle
The segment containing the minor arc is called the minor segment and the segment containing the major arc is called the major segment. The centre of the circle lies in the major segment.


Majorarc Segment of a Circle

Area of a Semi Circle

A diameter in a circle divides a circle into two equal arcs. Each of these two arcs is called a semi circle.
Area of Semicircle  
Area of a semi circle is half the area of the given circle.
Area of a semi circle = $\frac{(Area of the circle)}{2}$
                            = $\frac{(\pi r^{2})}{2}$
 

Properties of Circle

1. Circle The set of points which are at a constant distance of r units from a fixed point O and radius = r units.

2. Circumference: The perimeter or length of boundary of a circle is called its circumference

3. Radius: A line segment joining the centre of a circle and a point on the circle is called a radius of the circle.

4. Chord: A line segment joining any two points on a circle is called a chord of the circle

5. Diameter: A chord of a circle passing through its centre is called a diameter of a circle. It is the longest chord in a cirle

6. Secant: A line which intersects a circle at two points is called a secant of the circle

7. Arc: A continuous piece of a circle is called and arc of the circle. It consists of two endpoints and all the points between them.

8. Central Angle: An angle subtended by an arc at the centre of a circle is called its central angle

9. Semicircle: A diameter divides a circle into two  equal arcs. Each of these two arcs is called a semicircle

10. Segment: A segment of a circle is the region bounded by an arc and a chord, including the arc and the chord

11. Sector of a circle:  The region enclosed by an arc of a circle and its two bounding radii is called a sector of the circle

12. Quadrant:  One-fourth of a circular disc is called a quadrant. The central angle of a quadrant is 90°

13. Tangent:  A line meeting a circle only in one point is called a tangent to the circle at that point

14. The angle at the centre of a circle, subtended by an arc in a circle, is twice the angle at the circumference subtended by the same arc.

15. The angle in a semicircle is 90°, a right angle.