Areas Related to Circles Class 10 Notes CBSE Maths Chapter 12 (Free PDF Download)

Circles:

The path of a point moving in such a way that its distance from a fixed point is always the same is called a circle. That fixed point is called the centre of that circle and that path is called the locus of that point. The fixed distance between the centre and path is called the radius of that circle. We can see many examples of circles around us like bangles, round chapatis, dial watch, sun, etc.

• Perimeter of a circle – Perimeter is the circumferential length of a closed shape or a polygon. In case of circle, if we travel once around a circle, then the length covered gives us the perimeter of circumference. Circumference of a circle always bears a constant ratio with its diameter, which is denoted by a Greek letter $\pi $. Mathematically,

$\pi =\frac{Circumference}{Diameter}$

$\Rightarrow Circumference=\pi \times diameter$

$\Rightarrow Circumference=\pi \times d$

$\Rightarrow Circumference=\pi \times 2r$ (Where $r$ is the radius of circle and $d=2r$).

• Area of a circle – The space covered or occupied by a polygon in a two-dimensional plane is called the area. In case of a circle, it is the space occupied withing its boundary or the perimeter. If, $r$ is the radius of a given circle, then the formula for finding the area is given as;

$Area=\pi \times {{r}^{2}}$

Pi ($\pi $):

The value of $\pi $ was given by the great Indian mathematician Aryabhatta. He gave an approximate value of $\pi $ as $\pi =\frac{62832}{20000}$ which is almost equal to $3.1416$. It should be noted that $\pi $ is an irrational number as its value is non-terminating and non-recurring. For calculation purposes, we often take the value of $\pi $ as $\frac{22}{7}$ which in turn is a rational number.

Semicircle:

When a circle is cut into half along a diameter, semicircle is formed as shown below. Its perimeter consists of length of half a circle and the length of a diameter. If the semicircle is open, then diameter length is not added. If the length of diameter is given by $d$ and radius is given by $r$ then perimeter is given by,

$Perimeter=\pi r+d$ (For closed semicircle)

$Perimeter=\pi r$ (For open semicircle)

And the area of a semicircle is just half the area of a circle and is mathematically given as $\frac{\pi {{r}^{2}}}{2}$.

Similarly, area of a quadrant of a circle is given by $\frac{\pi {{r}^{2}}}{4}$.

Sector of a Circle:

The portion of a circle enclosed within an arc and two radii of that circle is called as sector.

Let us take the central angle between the radii is $\theta $ which is ${{360}^{\circ }}$ for a complete circle. Now let the length of that arc be $l$. Then the length $l$ can be found out using the following relation,

$l=\tfrac{\theta }{{{360}^{\circ }}}\times 2\pi r$.

Now, perimeter of sector is given as $2r+l$.

Similarly, area of sector is given by $\frac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}}$.

Segment of a Circle:

The part of the circular region enclosed between a chord and the corresponding arc of that circle is called the segment of a circle. The chord having centre of the circle as a point on it is the diameter and also the longest chord of the circle and divides the circle into two equal halves. When the chord is not the diameter, then the portion consisting the centre of circle is called the major segment and the other region is called the minor segment.

In the diagram above the chord, $BC$ divides the circle in two segments. Such as;

Area of minor segment$=$Area of sector $ABDC$$-$Area of $\Delta ABC$.

And area of major segment$=$Area of circle$-$Area of minor segment.

Here, area of $\Delta ABC$ can be found out using the formula $\frac{1}{2}{{r}^{2}}\sin \theta $.

And the area od sector $ABDC$ is given by $\frac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}}$.

Hence, the area of segment $ACB=\left( \frac{\theta }{{{360}^{\circ }}}\times \pi {{r}^{2}} \right)-\left( \frac{1}{2}{{r}^{2}}\sin \theta  \right)$

$={{r}^{2}}\left[ \frac{\pi \theta }{{{360}^{\circ }}}-\frac{\sin \theta }{2} \right]$.

Area of a Ring:

Ring is the region between two concentric circles having different radii. Let the radius of larger circle be $R$ and radius of smaller circle be $r$.

Hence the area of the ring is given by;

$\pi {{R}^{2}}-\pi {{r}^{2}}$

$=\pi \left( {{R}^{2}}-{{r}^{2}} \right)$.

Comprehensive Revision Notes for CBSE Class 10 Maths Chapter 12: Areas Related to Circles

Areas Related to Circles Class 10 Notes are prepared by Vedantu to help you revise your questions in this chapter. The following chapter presents several new concepts relating to a circle, for example, lines that cross the circle at different points forming components such as tangents, chords and diameters. This chapter helps you create a solid geometry basis for higher education and to achieve good results in the examinations. In real life also circles and their different properties, such as radius, diameter, circumference and area have applications.

So, the Introduction of Area Related to circles chapter is one of the important topics for Class 10 students from in higher studies point of view. Depending upon the properties & applications of the circles, few topics are designed in higher classes. So, let’s look into the important concepts of the circles which are discussed in this chapter:

• Introduction

• Area of a Circle

• Circumference of a Circle

• Segment of a Circle

• Minor arc and Major Arc

• Sector of a Circle

• Angle of a Sector

• Length of an arc of a sector

• Area of a Sector of a Circle

• Area of a Triangle

• Area of a Segment of a Circle

• Visualizations

• Areas of different plane figures

• Areas of Combination of Plane figures
  
  

Introduction

1. A circle is defined as a collection of points separated by a fixed distance, known as the radius, from a fixed point, known as the centre.

2. When a line and a circle are both in the same plane, the line and circle will not intersect. At a certain point, the line can come close to touching the circle. That kind of line is known as the tangent to the circle. The line is the secant for the circle as it intersects the circle at two points.
   
   

Tangent to a Circle

Tangent to a circle is the line that touches the circle at a single point. The point of tangency is the intersection of a tangent and a circle. The tangent is perpendicular to the circle's radius, in which it intersects. Any curved shape can have tangents. Since tangent is a line, it has its own equation.

• The tangent will touch the circle only at one point.

• We can name the line that contains the radius through a point of contact as ‘normal’ to circle at the point.
  
  

Condition of Tangency

The tangent is called only if it touches a curve at a single point. If not it is said to be simply called a line. So depending on the point of tangency, and also where it falls with respect to a circle, we can specify the criteria for tangent as follows:

• When the point lies inside of the circle.

• When the point lies on the circle.

• When the point lies outside of the circle.
  
  

Circumference of a Circle

The circumference of a circle, also known as its perimeter, is the measurement of the circle's boundary. The area of a circle, on the other hand, determines the region it occupies. The circumference of a circle is its length when we open it and draw a straight line through it. It's normally expressed in units like centimetres or metres.

⇒ Circumference (or) perimeter of a circle = 2πR

Area of a Circle

Area of a circle is nothing but the region occupied by the circle in a 2D plane. It can be determined by using the formula, A = πr2. (Here, r is the radius of a circle) This formula is useful while measuring the area occupied by a circular field or a plot.

Perimeter of Semicircle

The perimeter of a semicircle is nothing but the sum of half of the circumference of a circle and the diameter. We know that the perimeter of a circle is 2πr or πd. So, the perimeter of a semicircle will be ½ (πd) + d or πr + 2r, in which r is the radius.

Sector of a Circle

A sector is a section of a circle between its two radii and the adjacent arc. A semi-circle that represents half a circle is the most common sector of the circle. A circle that has a sector can be further divided into 2 regions called a Major Sector and a Minor Sector. You can find all the important topics explained in CBSE Class 10 Maths Notes Chapter 12 Areas Related to Circles PDF.

Benefits of Studying Vedantu’s Revision Notes:

Mathematics can be a difficult subject for Class 10 students to achieve good grades in, but if they prepare methodically by having revision notes, they can easily achieve more marks in their Maths exam.

• Areas Related to Circles Class 10 Notes will assist you in predicting the types of questions that could be asked during the examination.

• Solutions are split into various sections of the exam for a better understanding of the subjects.

• You can get a better understanding of the topics in simple language with our Revision notes.

• Solutions from Vedantu are error-free and well-organized.

• The questions are categorised such as short questions, long answer type questions, all the sections of the question paper in your school exams are thoroughly covered. If you solve these exercises extensively using Vedantu platforms as a reference source, you get full conceptual clarity in question and answer format. It is advisable to practise these questions because the activities in this chapter cover the course in-depth and are equally appropriate for quick review right before your exams.
  
  

Tips on How to Prepare for Exams Using Chapter 12 Areas Related to Circles

The tips given below will help students to prepare for their exams by using the free PDF of Areas Related to Circles Class 10 Notes available on Vedantu.

• Every question should be carefully read before attempting. Since there are some tough questions, there is a risk that we would give the incorrect answer if the questions are not completely understood.

• The basic formulas for finding circumference and area should be memorised in the circle chapter since they are fundamental formulas for solving any problems.

• Vedantu's Notes PDF includes several exercises and practise problems. To get good grades on your examinations, students can solve and practise these exercises several times.

• These solutions and concepts have been developed by Subject Experts to address your questions & doubts at the same time. This strategy will also allow you to increase your studying effectiveness in your self-study hours. For all your queries relating to 'Area Related to Circles,' Vedantu wants to provide you with a one-stop solution. These solutions are truly informative and provide you with realistic tips and tricks for correctly solving problems.
  
  

Conclusion

Vedantu's Areas Related to Circles Class 10 Notes for CBSE Maths Chapter 12 offer a comprehensive and valuable resource for students studying this topic. The free PDF download provided by Vedantu is a fantastic opportunity for learners to access high-quality study material without any financial burden. The notes cover essential concepts, formulas, and solved examples, enhancing students' understanding and problem-solving skills. With Vedantu's user-friendly approach, learners can grasp intricate concepts easily, fostering a deeper appreciation for the subject. Whether preparing for exams or seeking clarity on challenging topics, these notes serve as a reliable and effective aid, empowering students to excel in their academic journey.