Question

define circle.........??​

Answer 1

A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.

Answer 2

A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called “centre”. Every line that passes through the circle forms the line of reflection symmetry. Also, it has rotational symmetry around the centre for every angle. The circle formula in the plane is given as:

(x-h)2 + (y-k)2 = r2

where (x,y) are the coordinate points

(h,k) is the coordinate of the centre of a circle

and r is the radius of a circle.

How to Draw a Circle?

1. Take an empty sheet of paper and mark a single point on the sheet, somewhere in the middle of the sheet, and name it to point O.
2. Select a random length for radius, for example, 3 cm.
3. Using a ruler, keep the reference zero mark on point O and randomly mark 3 cm away from point O in all the direction.
4. Mark as many points as you want away from point O, but all of them should be exactly 3 cm away from point O.

Parts of Circle

A circle has different parts based on the positions and their properties. The different parts of a circle are explained below in detail.

Annulus-The region bounded by two concentric circles. It is basically a ring-shaped object.

Arc – It is basically the connected curve of a circle.

Sector – A region bounded by two radii and an arc.

Segment- A region bounded by a chord and an arc lying between the chord’s endpoints. It is to be noted that segments do not contain the centre.

Centre – It is the midpoint of a circle.

Chord- A line segment whose endpoints lie on the circle.

Diameter- A line segment having both the endpoints on the circle and is the largest chord of the circle.

Radius- A line segment connecting the centre of a circle to any point on the circle itself.

Secant- A straight line cutting the circle at two points. It is also called an extended chord.

Tangent- A coplanar straight line touching the circle at a single point.

Radius of Circle (r)

A line segment connecting the centre of a circle to any point on the circle itself “. The radius of the circle is denoted by “R” or “r”.

Diameter (d) of Circle

A line segment having both the endpoints on the circle. It is twice the length of radius i.e. d = 2r. From the diameter, the radius of the circle formula is obtained as r= d/2.

Circle Formulas

We know that a circle is a two-dimensional curve-shaped figure, and the two different parameters used to measure the circle are:

Area of  circle

Circumference of a circle

Circumference (C):- The circumference of a circle is defined as the distance around the circle. The word ‘perimeter’ is also sometimes used, although this usually refers to the distance around polygons, figures made up of the straight line segment. A circle circumference formula is given by

 C = πd = 2 π r

Where, π = 3.1415

Area (A):- Area of a circle is the amount of space occupied by the circle.The circle formula to find the area is given by

Area of a circle = πr2

Circle Area Proof

We know that Area is the space occupied by the circle.

Consider a concentric circle having an external circle radius to be ‘r.’

Open all the concentric circles to form a right-angled triangle.

The outer circle would form a line having length 2πr forming the base.

The height would be ‘r’

Therefore the area of the right-angled triangle formed would be equal to the area of a circle.

Area of a circle = Area of triangle = (1/2) ×b ×h

= (1/2) × 2π r  × r

Therefore, Area of a circle = πr2

Properties of Circles

The important basic properties of circles are as follows:

• The outer line of a circle is at equidistant from the centre.
• The diameter of the circle divides it into two equal parts.
• Circles which have equal radii are congruent to each other.
• Circles which are different in size or having different radii are similar.
• The diameter of the circle is the largest chord and is double the radius.

Some Question Regarding Circle :-

Example 1: -

Find the area and the circumference of a circle whose radius is 10 cm. (Take the value of π = 3.14)

Solution:

Given: Radius = 10 cm.

Area =π r2  

= 3.14  × 102

A= 314 cm2

Circumference, C = 2πr

C= 2 ×3.14×  10  

Circumference= 62.8 cm

Example 2:  -

Find the area of a circle whose circumference is 31.4 cm.

Solution:

Given:

Circumference = 31.4 cm

To find the area of a circle, we need to find the radius.

From the circumference, the radius can be calculated:

2 π r = 31.4

(2)(3.14)r = 31.4

r = 31.4 /(2)(3.14)

r=10/2  

r= 5

Therefore, the radius of the circle is 5 cm.

The area of a circle is πr2 square units

Now, substitute the radius value in the area of a circle formula, we get

A = π(5)2

A = 3.14 x 25

A =  78.5 cm2

Therefore, the area of a circle is 78.5 cm2.

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