Circles in Maths

In Maths or 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 centre 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 region. 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 precisely circular.

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

Circle Definition

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.

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. See the figure below.

• 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 the 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 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.

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

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

Let us discuss here the general formulas for area and perimeter/circumference of a circle.

Area and 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.

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