Circles

A circle is referred to as a round figure in a two-dimensional plane. Every point on the circle is equidistant from a particular point which is referred to as the center of the circle. The radius of a circle is a line segment connecting the centre of the circle to any point on the circle’s boundary. We generally use the small alphabet (r) to denote the length of the radius of a circle and (d) to represent the length of the diameter of a circle. Let us learn more about a circle, its properties, real-life examples, perimeter, and area with solved examples.

There are different properties associated with circles. It is important for students to understand these properties to be able to solve the sums on their own. Embibe offers students with PDF of NCERT books, previous year question papers and solution sets. Students can go through these study materials to understand the correct approach to answer the questions and improve their scores. Further, practicing from these study materials contribute towards exposing the students to a wider range of questions that will help them understand these concepts better.

Circle: Definition

A circle is the collection of all points in a plane at the same or equal distance from a fixed point. The fixed point is the centre of the circle and it is denoted by \(O\); the constant distance \(r\) from the centre to any point on the boundary corresponds to the radius of the circle.

Circle: Example

Objects which are round are called circular objects. There are many such circular objects that we see in our daily lives like a disc, basketball ring, tyres of a bicycle, bangles, base of the cylinder, the base of the cone, etc.

Circle: Properties

Radius

A line segment joining the centre to any point on the boundary of the circle is called the radius of the circle.

Diameter

A diameter of a circle is twice the length of a radius in a circle. It is a line segment that bisects the circle, so it is called a line of symmetry. It is the largest chord of a circle that passes through the circle’s centre.

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.

\({\rm{Diameter}}\,{\rm{of}}\,{\rm{the}}\,{\rm{Circle}}\,{\rm{ = }}\,{\rm{2}}\,\,{\rm{ \times }}\,{\rm{Radius}}\)

Tangent

A line that touches the circle at one point is called a tangent to the circle. The intersecting point at which the tangent touches the circle is called its point of contact.

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

Chord

When two distinct points on the boundary of the circle are joined then it is called a chord of the circle. A chord that passes through the centre of a circle is called the diameter of the circle. The circumference of a circle is its boundary.

Secant

A line that intersects the circle in two distinct points on the boundary of the circle is called a secant of the circle. It is the extension of a chord.

Segment

The region bounded by an arc of a circle and the chord of the circle whose endpoints are connected to the arc is called the segment of the circle.

If the chord is the diameter, it divides the circular region into two equal parts. But, if the chord is not the diameter, it divides the area into two unequal parts. The larger region contains the centre of the circle. This region is called the major segment, and the smaller region is called the minor segment of the circle.

Sector

The region bounded by an arc and the two radii joining the endpoints of the arc with the centre is called the sector of the circle.

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

Circle: Formulas

There are three important formulae that are related to the circles.

1. Diameter of a circle
2. Circumference of a circle
3. Area of a circle

Circumference of a Circle

Circumference of a circle or perimeter of a circle is the measurement of the length of the boundary of the circle.

Parts of the Circumference of a Circle

An arc when rotated about \({360^ \circ }\) forms the circumference of a circle.

If the arc is more than half of the circumference then it is a major arc. If it is less than half of the circumference it is a minor arc.

The diameter divides the circle precisely into two equal parts and passes through the centre.

Area of a Circle

The area of a circle is defined by the space or region occupied by a circle in a two-dimensional plane.

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

Area of a Circular Path

The circular path is the region between two concentric circles. If the radii of the outer circle and inner circle are r1 and r2 respectively,then

Breadth (or width) of the circular path = Outer radius – Inner radius

= r1 – r2

Area of circular path = Area of the bigger circle – Area of the smaller circle

=πr12-πr22=π(r12-r22)

Hence, the area of the circular path

=π(r12-r22)

Circles: Types

There are three different types of circles which are tangent circles, concentric circles, and congruent circles.

(i) Tangent Circles

Two or more circles that intersect at one and only one point. 

(ii) Concentric Circles

Two or more circles having the same centre but different radii are known as concentric circles.

Here are some more real-life examples of concentric circles:

(iii) Congruent Circles

Two circles are congruent if they are copies of one another or identical. That is, they have the same size.

Here are some more real-life examples of congruent circles:

Position of a Point

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., inside the circle (or interior of the circle), outside the circle (or exterior of the circle) and on the circle (or boundary of the circle)

Semicircle

When a circle is cut into two equal arcs along the diameter line then each of the two arcs is called a semicircle. A circle is divided into two equal semicircles by any diameter of the circle.

Quarter Circle

A quarter-circle is a quarter of a circle, formed by splitting a circle into four equal parts or a semicircle into two equal parts. A quarter-circle is also called a quadrant. 

Solved Examples

Question 1:The radii of the two circles are \(4\,{\rm{cm}},\,6\,{\rm{cm}}\) respectively. Find the length of their diameter.

Answer: 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 4 cm then the diameter is \(2 \times 4\;{\rm{cm}} = 8\;{\rm{cm}}\) and if the radius is 6 cm then the diameter is \(2 \times 6\;{\rm{cm}} = 12\;{\rm{cm}}.\)

Question 2: In the given figure, identify the major sector and the minor sector.

Answer: A sector is a part of a circle enclosing the region between the arc of the circle along with its two radii. It is a part of the circle which is formed by a part of the circumference (arc) and radii of the circle at both endpoints of the arc.

A sector divides the circle into two regions, namely the major and minor sectors. The smaller region is called the minor sector.

Hence, the minor sector is \(APB\) and the major sector is \(AQB\).

Question 3: Two diameters of a circle always intersect each other. State the reason.
Answer: The diameter of the circle is defined as the longest chord passing through the centre of a circle. In a circle an infinite number of diameters are possible and all of them pass through the centre. Hence, two diameters of a circle always intersect each other.

Question 4: I folded my circular disk into two parts and both the parts are equal. The equal parts are known as ______.
Answer: When a circle is cut into two equal arcs along the diameter line then each of the two arcs is called a semicircle. Hence, the answer is a semicircle.

Question 5:The ratio of the area of \(2\) circles is \(4:9\). Find the ratio of their radii.
Answer: 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 \( = {a_2}\)
It is given that \({a_1}:{a_2} = 4:9\)
The area of a circle \( = \pi {r^2}\)
\(\pi r_1^2:\pi r_2^2 = 4:9\)

Taking square roots on both sides,

\({r_1}:{r_2} = 2:3\)
Hence, the ratio of their radii is \(2:3\).

Summary

In this article, we have learned about the circle, its parts, formulas related to the circle, different parts or components of a circle. We have also studied the types of circles, their circumference, and their area. Practice other circle concepts with the given examples and Mock test questions in the article.

With the help of this article, one can also distinguish between the position of the points concerning a circle whether the point is inside, outside, or at the boundary of the circle. For applying the different concepts of circle discussed in this article some solved problems were also included.

The learning outcome of this article is that one can differentiate between the different parts of a circle and calculate the position of a point concerning a circle, its circumference, and area.

Related Topics to Study

FAQs

Below are the frequently asked questions on Circles:

Question-1: What are circles and their types?
Answer: Circles are round objects whose distance from a fixed point is the same. There are three types of circles concentric, tangent, and congruent.

Question-2: What are the two circles having the same centre and different radius called?
Answer: Circles that have the same centre and different radii are called concentric circles.

Question-3: What is the formula for circles?
Answer: Formulas related to circles are

Question-4: What is the centre of a circle called?
Answer: The centre of a circle is also called the focus of the circle.

Question-5: Does \(\pi \) radians represent a full circle?
Answer: No, \(\pi \) radians does not represent a full circle since in a half-circle there are \(\pi \) radians, which is equal to \({180^ \circ }\). Hence, π radians represent a semicircle. For a full circle, there are \(2\pi \) radians.

We hope this detailed article on circles is helpful to you. If you have any questions on circles, ping us through the comment box below and we will get back to you as soon as possible.