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Radius of a Circle | Definition, Formula | How to Find Radius
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Radius of Circle

Last Updated : 06 Jun, 2024
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Radius of Circle: The radius of a circle is the distance from the circle’s center to any point on its circumference. It is commonly represented by ‘R’ or ‘r’. The radius is crucial in nearly all circle-related formulas, as the area and circumference of a circle are also calculated using the radius.

In this article, we are going to learn about the Radius of the Circle in detail, including its Formula, Equation, and How to Find it with the help of Examples.

Radius of a Circle

Table of Content

What is the Radius of Circle?

Radius is a line segment that connects the center of a circle or sphere to its boundaries. The plural of radius is “radii”.

The diameter of a circle or sphere is the longest line segment connecting all points on the opposite sides of the centre, while the radius is half the length of the diameter.

Radius of a Circle Definition

The radius of a circle is the distance from the center of the circle to any point on its circumference. It is a constant length for a given circle and is half the diameter of the circle. The radius is typically denoted by the symbol r.

Diameter of Circle

Diameter is the line joining two points in a circle and passing through the centre of the circle. It is denoted by the symbol ‘d’ or ‘D’.

The diameter of the circle is twice its radius.

  • Diameter = 2 × Radius
  • Radius = Diameter/2

Diameter is the longest chord of the circle.

  • Circumference of Circle = πぱい(d)
  • Area of Circle = πぱい/4(d)2

Radius, Diameter and Chord

Any line passing through the circle can be categorized into three categories,

  • Secant to Circle
  • Tangent to Circle
  • Non-Intersecting Line

Illustration of Secant and Tangent

Secant to Circle

If a line touches the circle exactly two times then it is called Intersecting line. It is also called Secant to the circle.

Tangent to Circle

If a line touches the circle exactly one time then it is called a tangent to the circle.

Non-Intersecting Lines

If a line does not touch the circle then it is called Non-Intersecting Line.

  • Any line segment joining the centre of the circle to its circumference is called its radius.
  • A line segment joining two points on the circumference of the circle is called a chord of the circle.
  • The chord passing through the centre of the circle is called the diameter of the circle which is the longest chord of the circle.

Illustration of Radius, Diameter and Chord

Radius Formula 

Radius of a circle is calculated with some specific formulas which are given below in the table:

Formulas Related to Radius of Circle

Radius in Terms of Diameter d ⁄ 2
Radius in Terms of Circumference  C ⁄ 2πぱい
Radius in Terms of Area  √(A ⁄ πぱい)

where,

  • d is the Diameter of the Circle
  • C is the Circumference of the Circle
  • A is the Area of the Circle

How to Find Radius of Circle?

The radius of a circle can be found using the three basic radius formulas according to different conditions.

Let us use the following formulas to find the radius of a circle.

  • If the Diameter is known, Radius = Diameter / 2
  • If Circumference is known, Radius = Circumference / 2πぱい
  • If Area is known, Radius = √(Area of the circle/πぱい)

For example:

  • When the diameter is given as 28 cm, then the radius is R = 28/2 = 14 cm
  • When the circumference of a circle is given as 66 cm, then the radius is R = 66/2πぱい = 10.5 cm
  • When the area of a circle is given as 154 cm2, then the radius is R = √(154/πぱい) = 7 cm

Radius of Sphere

A sphere is a solid 3D shape. Radius of the Sphere is the distance between its centre and any point on its surface.

It can easily be calculated when the volume of the sphere or the surface area of the sphere is given.

Given Parameter Radius Formula
When Volume (V) is Given R = 3√{(3V) / 4πぱい} units V = Volume, πぱい ≈ 3.14
Surface Area (A) R = √(A / 4πぱい) units A = Surface Area, πぱい ≈ 3.14

Read More:

Radius of Circle Equation

Equation of circle on the cartesian plane with centre (h, k) is given as,

(x − h)2 + (y − k)2 = r2

Where (x, y) is the locus of any point on the circumference of the circle and ‘r’ is the radius of the circle.

If the origin (0,0) becomes the centre of the circle then its equation is given as x2 + y2 = r2, then Radius of Circle Formula is given by :

(Radius) r = √( x2 + y2 )

Chord of Circle Theorems

Theorem 1: Perpendicular line drawn from the centre of a circle to a chord bisects the chord.

Chord of Circle Theorem

Given: 

Chord AB and line segment OC is perpendicular to AB

To prove: 

AC = BC

Construction: 

Join radius OA and OB

Proof:

In ΔでるたOAC and ΔでるたOBC

∠OCA = ∠OCB (OC is perpendicular to AB)

OA = OB     (Radii of the same circle)

OC = OC     (Common Side)

So, by RHS congruence criterion ΔでるたOAC ≅ ΔでるたOBC

Thus, AC = CB (By CPCT)

Converse of the above theorem is also true.

Theorem 2: Line drawn through the centre of the circle to bisect a chord is perpendicular to the chord.

(For reference, see the Image used above.)

Given: 

C is the midpoint of the chord AB of the circle with the centre of the circle at O

To prove: 

OC is perpendicular to AB

Construction: 

Join radii OA and OB also join OC

Proof:

In ∆OAC and ∆OBC

AC = BC (Given)

OA = OB (Radii of the same circle)

OC = OC (Common)

By SSS congruency criterion ∆OAC ≅ ∆OBC 

∠1 = ∠2 (By CPCT)…(1)

∠1 + ∠2 = 180° (Linear pair angles)…(2)

Solving eq(1) and (2)

∠1 = ∠2 = 90° 

Thus, OC is perpendicular to AB.

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Radius of Circle Examples

Example 1: Calculate the radius of the circle whose diameter is 18 cm.

Solution: 

Given,

  • Diameter of the circle = d = 18 cm

Radius of the circle by using diameter,

Radius = (diameter ⁄ 2) = 18 ⁄ 2 cm = 9 cm

Hence, the radius of circle is 9 cm.

Example 2: Calculate the circle radius when circumference is 14 cm.

Solution:

Radius of a circle with a circumference of 14 cm can be calculated by using the formula, 

  • Radius = Circumference / 2πぱい

r = C / 2πぱい

r = 14 / 2πぱい {value of πぱい = 22/7}

r = (14 × 7) / (2 × 22)

r = 98 / 44

r = 2.22 cm

Therefore, the radius of the given circle is 2.22 cm

Example 3: Find the area and the circumference of a circle whose radius is 12 cm. (Take the value of πぱい = 3.14)

Solution: 

Given,

  • Radius = 12 cm

Area of Circle = πぱい r2 = 3.14  × (12)2

A = 452.6 cm2

Now Circumference of circle,

C = 2πぱいr

C = 2 × 3.14 × 12

Circumference = 75.36 cm

Therefore the area of circle is  452.6 cm2 and circumference of circle is 75.36 cm

Example 4: Find the diameter of a circle, given that area of a circle, is equal to twice its circumference.

Given,

  • Area of Circle = 2 × Circumference

We Know, 

  • Area of the circle = πぱい r2
  • Circumference = 2πぱいr

Therefore,

πぱい r2 = 2×2×πぱい×r

r = 4

Therefore,

diameter = 2 × radius

diameter = 2 × 4 = 8 units

Practice Questions on Radius of Circle

Q1. What is the Radius of circle if its Area is 254 cm2?

Q2. Find the area of circle with circumference 126 units.

Q3. Find the diameter of the circle if its radius is 22 cm.

Q4. Find the area of the circle with diameter 10 cm.

FAQs on Radius of Circle

Define Radius of Circle.

The line joining the centre of the circle to any point in its circumference is called the radius of the circle. It is denoted by ‘r’ or ‘R’

How Many Radii can be drawn in Circle?

A circle can have infinite radii drawn inside it.

What is the Radius of Unit Circle?

A unit circle is a circle with a radius 1 unit.

What is the Relation between Radius and Diameter of Circle?

Diameter of a circle is twice the radius of the circle.  Diameter = 2 × radius

How to Find Radius of Circle?

Radius of a circle is the found using various formulas that are,

  • If the Diameter is known. Radius = Diameter / 2
  • If Circumference is known. Radius = Circumference / 2πぱい
  • If Area is known. Radius = √(Area of the circle/πぱい)

How to Find the Radius of Circle with Area?

To find the Radius of a Circle when Area is given, we use the following formula :

Radius = √(Area of the circle/πぱい)

How to Find the Radius of Circle with Circumference?

To find the Radius of a Circle when Circumference is given, we use the following formula :

Radius = Circumference / 2πぱい.



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