Radius of Circle (original) (raw)

Last Updated : 23 Jul, 2025

**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

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 is the longest chord of the circle.

Radius, Diameter and Chord

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

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.

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
Radius in Terms of Area √(A ⁄ π)

where,

**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.

**For example:

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} units V = Volume, π ≈ 3.14
Surface Area (A) **R = √(A / 4π) units A = Surface Area, π ≈ 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 = r 2

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 = √( x** 2 + y 2 )

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 ΔOAC and ΔOBC

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

OA = OB (Radii of the same circle)

OC = OC (Common Side)

So, by RHS congruence criterion ΔOAC ≅ ΔOBC

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,

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,

r = C / 2π

r = 14 / 2π {value of π = 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 = 3.14)

**Solution:

Given,

Area of Circle = π r2 = 3.14 × (12)2

A = 452.6 cm2

Now Circumference of circle,

C = 2πr

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,

We Know,

Therefore,

π r2 = 2×2×π×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 cm 2 ?

**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.