Radius of a Circle

Dhristi JEE 2022-24

Radius of a Circle Definition

When you move around with respect to a specific point, then it forms a circle, only if you move in the fixed path. The point which you are taking as your references is called the centre of the circle. The path you follow while moving around forms the circumference of the circle. The distance that remains fixed while moving about a point is called the radius of a circle. Working with circles has always been very interesting. It is an important part of the mathematics concept to study.

 

The radius of a circle is the distance from the centre of the circle to any point on its circumference. It is usually denoted by ‘R’ or ‘r’. The area and circumference of a circle are also measured in terms of radius.


A Radius can be defined as a measure of distance from the centre of any circular object to its outermost edge or boundary. It is not only a dimension of a circle but also the dimension for a sphere, a semi-sphere, a cone with a circular base and a cylinder having circular bases. 

 

A Circle can be defined as the locus of a point moving in a plane, in such a manner that its distance from a fixed point is always constant, and this fixed point is known as the centre of the circle and the distance between any point on the circle and its centre is the radius of a circle.

 

The Diameter of a Circle is the length of the line that starts from one point on a circle to another point and passes through the centre of the circle, and it is equal to twice the radius of the circle. It is denoted by ‘d’ or ‘D’.

Diameter = 2 x Radius

Or 

Radius = Diameter/2.

 

The diameter is the longest chord of the circle. 

 

We can also express the area and circumference of a circle with respect to the diameter.

 

Here, the Circumference of circle = π (Diameter)

 

Area of circle = π/4 (Diameter)2.

 

Definition of the Radius of a Circle and the Chord

A chord is the line segment that joins two different points of a circle that can also pass through the centre of the circle. If a chord passes through the centre of the circle, it is known as the diameter. The radius of a circle refers to any line segment that connects the centre of the circle to any point on the circle. The chord of a circle refers to the line segment that joins any two points on the circle.


Length of Chord of Circle Formula

There are two different formulas for calculating the length of the chord of a circle. The formulas are: 


Length of the chord = 2 × √(r2 – d2). This formula is used when calculated using a perpendicular that is drawn from the centre. 


For use in Trigonometry, the Length of the chord = 2 × r × sin(c/2), where r is the radius, d is the diameter, and c will be the centre angle subtended by the chord. 


What is a Circle?


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This diagram shows a circle with centre at O and radius being the same to all points on the circumference.

 

Define Radius of a Circle

According to classical geometry, the radius of a circle is defined as the equal distance drawn from the centre to the circumference of the circle. If we double this distance, then it becomes the diameter of the circle.

 

Define Relation Between Radius of a Circle and Chord

A chord is the line segment that joins two different points of the circle which can also pass through the centre of the circle. If a chord passes through the centre of the circle, then it becomes diameter. 

Suppose, here we consider d as the diameter, then the radius is given by 

d = r/2

The diameter of the circle is the longest chord.

Let us describe the concept of a chord with the help of a diagram.

 

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The above diagram represents a line segment that intersects the circle at A and B.

AB is the chord of the circle in the above diagram. If this AB passes through the centre at O, then it becomes diameter which is two times of the radius.

 

Chord of a Circle Theorems

Theorem 1: 

The line drawn to the chord from the centre bisects it at the right angle.

 

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In the above diagram, AB is the chord and OC is drawn from the centre to point C at AB. We have to prove if AC= BC

 

Solution:

Form the triangles drawing AO and OB. 

According to the statements, 

AO=OB

OC is common for both the triangles, angle OCA = angles OCB = 90-degree

Hence the two triangles are congruent to each other. 

So, AC = BC

 

Theorem 2: 

To prove that a line bisecting the chord of a circle drawn from the centre is perpendicular to the chord.

 

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In the above diagram,

AC = CB

We have to prove OC is perpendicular to AB.

Form two triangles by joining OA and OB

In the two triangles AOC and BOC, 

OA = OB

AC = BC 

OC= OC (common to both)

Hence the two triangles are congruent to each other by SSS property. 

According to the linear pair

Angle 1 + angle 2 = 180

Also, angle 1 = angle 2

Hence angle 1= angle 2 = 90-degree

Hence proved OC is perpendicular to AB.

 

Length of Chord of Circle Formula

We have two different formulas to calculate the length of the chord of a circle. Below are the mentioned formulas. 

Length of the chord = 2 × √(r2 – d2)

This formula is used when calculated using a perpendicular drawn from the centre.

If you are using trigonometry, 

Length of the chord = 2 × r × sin(c/2)

Here r will be the radius, d is the diameter, and c will be the centre angle subtended by the chord.

 

What is an Arc and Chord of a Circle?

An arc is the part of the circumference of the circle. It is the curved part of the circle. However, a chord will be the line segment drawn by the two different points on the circle. A sector helps in finding the length of the arc. 

A sector is the portion of the circle formed by two radii of the circle. Below are the given descriptions to each with the help of the diagram. 

 

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The above diagram describes the sector formed by two radii OB and OA. 

The above diagrams form an arc AB formed by joining two radii OA and OB.

The above shows a chord GH which is a line segment formed by joining points G and H.

FAQs on Radius of a Circle

1. What are the Different Properties of a Circle?

Below are the mentioned properties of the circle:

  • Two circles are congruent if they have the same radii. 

  • Diameter is the most extended chord that passes through the centre of the circle.

  • If the radius and chord of two circles are equal, then they have equal circumference. 

  • The two-chord theorems are valid for all the circles. 

  • You can inscribe a circle inside other geometrical shapes. 

  • The two chords which are equidistant from the centre of the same circle are the same in length.

  • If you draw a line from the centre of the circle to the diameter, the distance is zero. 

  • If the length of the chord drawn to a circle increases, the length of the perpendicular distance decreases. 

These are some mentioned properties of the circle which are valid universally. 

2. Which Statement Contradicts the Properties of the Circle?

In a circle, it is defined as a closed figure as other geometrical shapes and figures. However, the statement itself is contradictory. A circle has an inside and an outside area just like other geometric shapes. However, it is contradictory in the fact that other shapes and figures like square, rectangle, triangle, and trapezium have some angles or straight sides. However, there are no such conditions in the case of the circle. It does not form any angle or the straight line. Thus the statement and definition of the circle are itself contradictory to the properties of a closed figure.

3. What is an Arc and Chord of a Circle?

An arc is a part of the circumference of the circle. It is part of a curve of a circle. A chord is the line segment drawn by the two different points on the circle. It is a straight line with both endpoints on the circle. The diameter is the longest chord of a circle that passes through the centre of the circle. A sector is the portion of the circle formed by two radii of the circle and a sector helps in finding the length of the arc.   

4. Mention some properties of a Circle?

Properties of a Circle are:


(i) Circles are said to be congruently provided they have an equal radius;


(ii) The longest chord of a circle is the Diameter;


(iii) When the Radius and the Chord of two Circles are equal, they then have an equal circumference;


(iv) The Radius, when drawn perpendicular to the Chord, bisects the Chord;


(v) The Two chord theorems apply to all circles.


(vi) A circle can be inscribed inside various other geometrical shapes.

5. Which definition contradicts the properties of a Circle?

A Circle is defined as a closed figure as the other geometrical shapes and figures. But, as we know, a Circle consists of an inside as well as an outside area like other geometric shapes. However, the above definition is contradictory as the other shapes and figures like the square, the rectangle, the triangle, and the trapezium, all have some angles or straight sides. In a circle, however, there is no such property. It does not form any angle or a straight line.

6.  What are the chord properties of Circle?

The properties of a circle that are related to chord are:


(i) A perpendicular line drawn from the centre of the circle to the chord always bisects the chord. 


(ii) The line that is drawn through the centre of the circle so as to bisect the chord is always perpendicular to the chord.


(iii) Equal chords of a Circle are known to subtend equal angles at the centre.

7. Where can I get Study Materials on the Radius of a Circle?

In Mathematics, the Radius of a Circle is an important chapter and it is necessary to be able to practice some of the important questions to be able to do well. The online portal, Vedantu offers important questions along with answers and other very helpful study material on Radius of a Circle - Definitions, Theorems and Length of Chord, which have been formulated in a  well structured, well researched, and easy to understand manner. These study materials and solutions are all important and are very easily accessible from Vedantu.com and can be downloaded for free. 

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