Radius of a Circle: Definition, Formula & Examples

Radius of a circle is any line from the center of the circle to the circumference of the circle. A diameter is the sum of two radius within the circle. Any line apart from the diameter inside the circle is referred to as a chord. The radius of a circle is used for the purpose of calculating the area and circumference of the circle. Students need to understand the basics related to a circle to be able to solve the problem sums associated with the same. This article will focus on defining the circle and will further discuss the formulas with relevant examples for the convenience of the students to learn these concepts with utmost ease.

Circle is an important chapter in Mathematics CBSE 10th board. For the academic session of 2021-22 NCERT has decided to conduct the board examination in two terms. There is no official notification from NCERT on the exam dates of term- 2 examinations. Students can follow the Embibe page to receive all the information updated by NCERT. Embibe further offers solution sets and MCQ mock test papers that will help students to revise and practice all the sums appropriately. Students can follow these study materials to enhance their preparation for board examinations.

Circle: Definition

A circle can be defined as a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a fixed point called the center. There is only one center in every circle. The distance from the center to the edge of the circle is always the same.

Circle: Properties

There are different aspects associated with the circle that needs to be understood to in order to be able to solve the problem sums correctly.

Circle: Chord

A straight line joining any two points on the circumference of a circle is called a chord. We can say that when two distinct points on the boundary of the circle are joined, then the line segment formed is called a chord of the circle. The circumference of a circle is its boundary.

Circle: Diameter

A chord of a circle that has the center in it is referred to as the diameter of a circle. Diameter is the largest chord of a circle that passes through the center of the circle.

There is a circle with \(O\) as a centre and \(OP\) as a radius. In the figure, you can see the line segment \(AOB\) that passes through the centre of the circle \(O\) and has its endpoints at the circumference. \(AOB\) is the diameter of the circle.

A diameter of a circle is twice the length of a radius in a circle.

Diameter of the Circle \( = 2 \times \)Radius

Circle: Radius

The radius of a circle is the distance between the center of the circle to its circumference. The radius is exactly half of the diameter. The formula of the radius can be simply derived by dividing the diameter of the circle by two.

Radius \({\rm{ = }}\frac{{{\rm{Diameter}}}}{{\rm{2}}}\)

Circle: Tangent

Any straight line touching a exterior of a circle is referred to as a tangent to a circle. The point of contact is the intersecting point at which the tangent touches the circle. In the figure given below, the point \(P,\) at which the tangent is touching the circle, is the point of contact.

In the given figure, SPT is the tangent to the circle at point \(P.\)

As now we are thorough with the definition of a tangent, here is a quick fact about the tangent and radius of a circle.

The radius of a circle is perpendicular to the tangent. Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

Circle: Secant

A straight line intersecting a circle at two points is called a secant. In the given figure, line \(FG\) intersects the given circle at point \(P\) and \(Q.\) Therefore, \(FG\) is a secant.

If the secant passes through the centre of the circle, then the secant contains the diameter and radius of the circle.

Circle: Segment

Every chord divides a circle into two parts; these two parts of a circle are called its segments. Therefore, a segment is a part of a circle enclosed by a chord and an arc.

In the given figure, the chord \(AB\) divides the circle into two unequal parts.

The smaller, i.e., the coloured part is called the minor segment, and the larger, i.e., unshaded part of the circle, is called the major segment.

Circle: Sector

The part of a circle enclosed by any two other radii and an arc is called a circle.

When a major arc forms a sector, it is called the major sector, and when the sector is formed by a minor arc, it is called the minor sector.

Circle: Radius

The radius is the distance from the centre outwards. The diameter goes straight across the circle, through the centre. The diameter is double the radius. Hence, the radius is found by dividing the diameter by \(2.\)

Position of a Point with Respect to a Circle

Let us identify the position of a point with the help of a radius.

The part of the plane or the set of points in a plane that lies inside a circle is called the interior of the circle. The part of the plane or the set of points in a plane that lies outside a circle is called the exterior of the circle. The set of points that lies on the circumference of the circle is called on the boundary of the circle.

Consider a circle in a plane, as shown in the figure. Consider any point \(P\) on the circle. If the distance from the centre \(O\) to the point \(P\) is \(OP,\) then

(i) \(OP=\) radius (If the point \(P\) lies on the circle)                                     

(ii) \(OP<\) radius (If the point \(P\) point lies inside the circle) 

(iii) \(OP>\) radius (If the point \(P\) lies outside the circle) 

Therefore, a circle divides the plane into three parts, i.e., the interior of the circle, the exterior of the circle, and the circle’s boundary.

Radius of a Circle: Application

The function of a radius of a circle is used to find the diameter, circumference, and area of a circle. Let’s discuss it one by one.

1. Radius to Find the Diameter: As already discussed, the diameter of a circle is twice the length of a radius in a circle.

And thus, the diameter of a circle \( = 2 \times \)Radius.

2. Radius to find the Circumference: Circumference of a circle or perimeter of a circle is the measurement of the length of the boundary of the circle. 

If the radius of a circle is known, then the circumference of the circle can be calculated by using the formula,

Circumference \( = 2\,\pi r\)

where \(r\) represents the radius of the circle and is the mathematical constant whose value is equal to \(\frac{{22}}{7}\) or \(3.14.\) (it is an irrational number with non-terminating decimal places)

3. Radius to find the Area: We know that the area can be defined as the space occupied by a flat shape or the surface of an object. Here, the area of a circle is a region occupied by the circle in a \(2-D\) plane.

If \(r\) be the radius of a circle, then the area of a circle is given by \(\pi {r^2}\).

Solved Examples – Radius of a Circle

Q.1. The radii of the two circles are \({\rm{10}}\,{\rm{cm, 15}}\,{\rm{cm}}\) respectively. Find the length of their diameter.
Ans: The radius is half of the diameter. It starts from a point on the boundary of the circle and ends at the centre of the circle.
Thus, the diameter is double of a radius.
Therefore, if the radius is \({\rm{10}}\,{\rm{cm,}}\) then the diameter is \(2 \times 10\;{\rm{cm}} = 20\;{\rm{cm}},\) and if the radius is \({\rm{15}}\,{\rm{cm,}}\) then the diameter is \(2 \times 15\;{\rm{cm}} = 30\;{\rm{cm}}.\)

Q.2. Given the radius of a circle is \(r\,{\rm{cm}}\) and if it is doubled, then what will be the circumference of the new circle.
Ans: Given, the radius of the circle \( = r\,{\rm{cm}}\)
Then the circumference of the circle \( = 2\pi r\)
If the radius of the circle is doubled then, new radius \(R = 2r\;{\rm{cm}}\)
Therefore, the circumference of the new circle \( = 2\pi R = 2\pi \times 2r = 4\pi r\)
Hence, the circumference of the new circle is \(4\pi r.\)

Q.3. The ratio of the area of two circles is \(16:25.\) Find the ratio of their radii.
Ans: Let the radius of the first circle \( = {r_1}\)
Area of the first circle \( = {a_1}\)
The radius of the second circle \( = {r_2}\)
Area of the second circle \( = {r_2}\)
It is given that \({a_1}:{a_2} = 16:25\)
The area of a circle \( = \pi {r^2}\)
\(\pi r_1^2:\pi r_2^2 = 16:25\)
Taking square roots on both sides,
\({r_1}:{r_2} = 4:5\)
Hence, the ratio of their radii is \(4:5.\)

Q.4. The radius of two circles is in the ratio \(1:x\) What is the ratio of their areas?
Ans: Given \(\frac{{{r_1}}}{{{r_2}}} = \frac{1}{x}\)
So, \(\frac{{{A_1}}}{{{A_2}}} = \frac{{\pi \times r_1^2}}{{\pi \times r_2^2}} = \frac{1}{{{x^2}}}\)
Hence, the ratio of their areas \(1:{x^2}\).

Q.5. The circumference of a circle exceeds the diameter by \({\rm{40}}\,{\rm{cm}}\) Find the radius of the circle. Take \(\pi = \frac{{22}}{7}.\)
Answer: Let the radius of the circle \( = r\,{\rm{cm}}\)
Then, the circumference of a circle \( = 2\,\pi r\)
Since circumference exceeds diameter by \({\rm{40}}\,{\rm{cm}}\) Therefore, according to the question,
\(2\pi r = d + 40\)
⇒ \( \Rightarrow 2\pi r = 2r + 40\)
\( \Rightarrow 2 \times \left( {\frac{{22}}{7}} \right) \times r = 2r + 40\)
\( \Rightarrow \frac{{44r}}{7} – 2r = 40\)
\( \Rightarrow \frac{{(44r – 14r)}}{7} = 40\)
\( \Rightarrow \frac{{30r}}{7} = 40\)
⇒ \( \Rightarrow r = \frac{{7 \times 40}}{{30}}\)
⇒ \( \Rightarrow r = \frac{{28}}{3}\;{\rm{cm}}\)
Hence, the radius of the circle is \(\frac{{28}}{3}\;{\rm{cm}}{\rm{.}}\)

Summary

In this article, we learned about the definition of the radius of a circle. We also familiarized ourselves with the concept that diameter is double the radius. In addition, we also learned about the different parts of a circle and how to find the diameter, circumference and area of a circle with the help of the radius.

Frequently Asked Questions (FAQs) – Radius of a Circle

Frequently asked questions related to radius of a circle is listed as follows:

Q.1. What are chords?
Ans: A straight line joining any two points on the circumference of a circle is called a chord.

Q.2. What is \({\rm{2 \times radius}}\) called?
Ans: A diameter of a circle is twice the length of a radius in a circle. Thus, \({\rm{2 \times radius}}\) is called the diameter of a circle.

Q.3. What is the radius of a circle in math?
Ans: A line segment joining the centre to any point on the boundary of the circle is called the radius of the circle.

Q.4. How to find the radius of a circle if the circumference is given?
Ans: The circumference of a circle and radius are related to each other, and their relation can be expressed as \(C = 2\pi R\) So, if the circumference is known, the radius of the circle is \(r = \frac{C}{{2\pi }}\)

Q.5. What are radius and diameter?
Ans: The radius of a circle is the distance from the centre of the circle to any point on its circumference. A diameter of a circle is twice the length of a radius in a circle. It is the largest chord of a circle that passes through the centre of the circle. It is a line segment that bisects the circle.

We hope you find this article on Radius of a Circle helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.