Radius of a Circle

Radius of a Circle Definition

When you move around with respect to a specific point, then it forms a circle, only if you moved 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.


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 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 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 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. 

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

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

FAQs (Frequently Asked Questions)

Q1. 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 drew 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. 

Q2. 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.