Radius of a Circle

Last updated on 15 December 2024

Table of contents

Introduction to the Radius of Circle
Definition of Radius
How to calculate radius?
Conclusion
FAQs on Radius of Circle

Introduction to the Radius of Circle

The radius of a circle is one of the most fundamental concepts in Geometry, serving as a bridge to understanding many other mathematical concepts. Whether you are exploring the beauty of circular shapes in everyday life or delving into the intricacies of geometric calculations, the radius plays a central role. It is the simple yet powerful measure from the center of a circle to any point on its circumference, forming the basis for calculating diameter, circumference, and area of the circle.

Definition of Radius

A line segment from the centre of the circle to any point on its circumference is called the radius of the circle. In other words, radius is the distance from the center of a circle to any point on its circumference.

This means that a circle has an infinite number of radii because there are infinite points on its circumference.

Radius is denoted by "r" or "R".
The plural of the radius is "radii".
It is half of the length of the diameter.
The size of the circle changes when the length of the radius varies.
All the radii of the circle have the same length.

In the figure above, points A, B, X, Y, P, Q, M and N lie on the circumference of the circle with centre O. These points are equidistant from the centre O. So, the line segments OA, OB, OX, OY, OP, OQ, OM and ON are radii of the circle and all these radii are equal.

OA = OB = OM = ON = OP = OQ = OX = O Y

How to calculate radius?

In this section, let's learn to find the length of the radius of a circle. The radius of a circle can be calculated using certain formulas based on what details are given.

Calculating Radius Using Diameter

Radius is half the length of a diameter. Hence if the diameter of a circle is given the radius formula is expressed as

r = \frac{D}{2}

where

D

is the length of the diameter of the circle.

Example: If the diameter of a circle is

8 cm

, calculate its radius.

Given: $D = 8 cm$

To find: radius of the circle

Solution:

\begin{array}{lrcl} ∵ & r & = & \frac{D}{2} \\ ∴ & r & = & \frac{8}{2} \\ ∴ & r & = & 4 cm \end{array}

Therefore, the radius of the given circle is

4 cm

Exercise:

Find the radius of a circle if its diameter is 12 cm.
Find the radius of a bike tyre whose diameter is 22 inches.

Calculating Radius Using Circumference

Circumference is the total length of the boundary of the circle and its formula is expressed as:

C = 2 π r

where

r

is the radius of the circle and

π \approx 3.14

If circumference is given, above formula can be used to calculate the radius of the circle.

Example: Find the radius of a circle if its circumference is

18 cm

Given: $C = 18 cm$

To find: radius of the circle

Solution:

\begin{array}{lrcl} ∵ & C & = & 2 π r \\ ∴ & \frac{C}{2 π} & = & r \\ ∴ & r & = & \frac{C}{2 π} \\ ∴ & r & = & \frac{18}{2 \times 3.14} \\ ∴ & r & = & \frac{9}{3.14} \\ ∴ & r & = & 2.8662 cm \approx 2.87 cm \end{array}

Therefore, the radius of the given circle is

2.87 cm

Exercise:

Find the radius of a circle if its circumference is 15 mm.
Find the radius of a circular clock whose circumference is 38 inches.

Calculating Radius Using Area of a Circle

Area of a circle is defined as the region occupied by the circle in a two-dimensional plane. It is determined by the formula below.

A = π r^{2}

where

r

is the radius of the circle and

π \approx 3.14

If area of a circle is given, above formula can be used to calculate the radius of the circle.

Example: If the area of the circle is

314 {cm}^{2}

, find its radius.

Given: $A = 314 {cm}^{2}$

To find: radius of the circle

Solution:

\begin{array}{lrcl} ∵ & A & = & π r^{2} \\ ∴ & \frac{A}{π} & = & r^{2} \\ ∴ & r^{2} & = & \frac{A}{π} \\ ∴ & r & = & \sqrt{\frac{A}{π}} \\ ∴ & r & = & \sqrt{\frac{314}{3.14}} \\ ∴ & r & = & \sqrt{100} \\ ∴ & r & = & 10 cm \end{array}

Therefore, the radius of the given circle is

10 cm

Exercise:

If the area of the circle is $157 {km}^{2}$ , find its radius.
Area of the circular saucer is $50.24 {cm}^{2}$ , find its radius.

Conclusion The radius of a circle serves as a fundamental element in geometry, mathematics, and various fields of science and design. Its simplicity and profound significance, impacts everything from architectural wonders to the aesthetics of art. As we delve into the details of the radius, we discover not only its mathematical properties but also its intricate connections to the world around us, making it a cornerstone in the rich tapestry of mathematical understanding.

FAQs on Radius of Circle

What is the radius of a circle?
The radius is the distance from the center of the circle to any point on its circumference.
How is the radius different from the diameter?
The diameter is twice the length of the radius and stretches across the circle, passing through the center of the circle.
How do you find the radius of a circle if the diameter is given?
Divide the diameter by 2 to find the radius of the circle.
$Radius = \frac{Diameter}{2}$
Can the radius be a negative value?
No, the radius represents a distance, and distances are always positive.

Radius of a Circle

Introduction to the Radius of Circle

Definition of Radius

How to calculate radius?

Calculating Radius Using Diameter

Calculating Radius Using Circumference

Calculating Radius Using Area of a Circle

FAQs on Radius of Circle

What is the radius of a circle?

How is the radius different from the diameter?

How do you find the radius of a circle if the diameter is given?

Can the radius be a negative value?

Circle

Diameter of a Circle