Circles

In Maths or in Geometry a circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident. A circle is also termed as the locus of the points drawn at an equidistant from the centre. The distance from the center of the circle to the outer line is its radius. Diameter is the line which divides the circle into two equal parts and is also equal to twice of the radius.

A circle is a basic 2D shape which is measured in terms of its radius. The circles divide the plane into two regions such as interior and exterior regions. It is similar to the type of line segment. Imagine that the line segment is bent around till its ends join. Arrange the loop until it is exactly circular in shape.

The circle is a two-dimensional figure, which has its area and perimeter. The perimeter of circle is called circumference. The area of the circle is the region bounded by it in a 2D plane. Let us dicuss here about circle definition, formulas, important terms with examples in detail.

Circle Definition in Maths

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

In maths projects for class 10 on circles, the construction of a circle and all the properties and terminologies are explained in detail. To understand what circles are in simple terms, go through circles for class 10, and also try the following exercise –

Circles

  1. Take an empty sheet of paper and just 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 3cm away from point O in all the direction.
  4. Mark as many points as u want away from point O, but all of them should be exactly 3 cm away from point O.

If you’ve selected sufficient points, you may notice that the shape is starting to resemble a circle and this is exactly what the definition of a circle is.

Circles Terminologies

  • Annulus-The region bounded by two concentric circles. It is basically a ring-shaped object. See the figure below.

Annulus of a circle

  • 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.

See the figure below explaining arc, sector and segment of a circle.

Arc, sector and segment of a circle

  • 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
  • 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 as an extended chord.
  • Tangent- A coplanar straight line touching the circle at a single point.

See the figure below-representing the centre, chord, diameter, radius, secant and tangent of a circle.

Circles perimeterRadius 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”.

Radius of a Circle

Diameter (d) of Circle

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

Diameter of a CircleAlso, read:

Circle Formulas for Area and Circumference

Let us discuss here the general formulas for carea and perimeter/circumference of 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

Circumference of a Circle
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

Area of a Circle

Circle Area Proof

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

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

Area of a Circle

Open all the concentric circle 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

Here are some basic properties of circles.

  • The outer line of a circle is at equidistant from the center.
  • 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 of the radius.

Circles Example Problem

Example- 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 rSquare units

= 3.14  × 102

A= 314 cm2

Circumference, C = 2πr units

C= 2 ×3.14×  10 

Circumference= 62.8 cm

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