Class 10 Maths Areas Related to Circles Most Important Questions with Solutions

Circles are everywhere, from the wheels on your bike to the orbit of planets in the solar system. Their geometry is used in engineering, architecture, design, and even in technology like satellites and robotics. The reason for their importance is not just because they’re common shapes, but because of their unique mathematical properties.

When studying geometry, one of the most interesting topics you will see is the circle. For Class 10 students, the chapter on "Areas Related to Circles" plays a very important role in developing a deeper understanding of geometry. 

Educart has come up with Areas Related to Circles Class 10 Important Questions explaining the importance of these concepts, why using them is important mastering them is essential, and how they can be used effectively.

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Chapter 11 Areas Related to Circles: Important Questions

1. Shown below are two pendulums of different lengths attached to a bar.

(Note: The figure is not to scale.)

Based on the figure shown above, the arc length of pendulum 1 is length of pendulum 2. a. greater than

b.  lesser than

c. equal to

d. (cannot be answered without knowing the value of R.)

Soln.

(c) equal to

Explanation:

Arc length =θ/360°×2πr

Pendulum 1 arc length:

⇒60°/360°×2π×3/4R

⇒1/6×π×3/2R

⇒π/4R

Pendulum 2 arc length:

⇒45°/360°×2πR

⇒18×2πR

⇒π/4R

Both the arc length are equal.

∴ The arc length of pendulum 1 is  equal to the arc length of pendulum 2.

2. The figure below is a part of a circle with centre O. Its area is 1250π/9 cm2 and the 10 sectors are identical. 

(Note: The figure is not to scale.)

Find the value of ß, in degrees. Show your steps.

Soln.

Given, area =1250π/9cm²

We know that Area of a sector=θ/360°×πr²

1250π/9=θ/360×π×20²

1250/9=

θ/360×20² (1250×360/9×400)=θ

⇒θ=125°

⇒β=125/10

=12.5°

Hence, the required answer is 12.5°

3. A cow is tied at one of the corners of a square shed. The length of the rope is 22 m. The cow can only eat the grass outside the shed as shown below.

(Note: The figure is not to scale.)

What is the area that the cow can graze on? Show your steps.

(Note: Give the answer in terms of π.)

Soln. 

Area that the cow can graze on =3quarter sector with radius 22m +2one quarter sector with radius 2m

⇒3/4×π×22²+2/4×π×2²

⇒363π+2π

⇒365π m²

4. Ramit drew two circles of different radii. Each of them had an arc that subtended an equal angle at the centre.

He said, "Both arcs are of the same length".

i) Is Ramit right?

ii) If both radii and angles subtended by the two arcs are different, can the arc lengths be the same?Give valid reasons.

Soln. 

Arc length =x/360°×circumference of circle

i) Given that two circles are drawn with different radii and same central angle. So, the arc lengths of these two circles will also be different. 

Therefore, Ramit is not right as the arc length depends on the radius of the circle. 

ii) If both radii and angles subtended by the two arcs are different, the arc lengths can be same when the product of measure of the angle and radius is the same for both circles.

5. Shown below are two overlapping sectors of a circle. The radii of the sectors are 6 cm and 8 cm. The figure is divided into three regions - I, II and III.

(Note: The figure is not to scale.)

Find the difference in the areas of regions I and III. Show your work.

(Note: Take π = 22/7)

Soln. 

Radii of sectors= 6 cm and 8 cm

Therefore, we get:

The area of the sector with radius 8 cm=

60/360×π×8²

= 64π/6 cm²

The area of the sector with radius 6 cm= 60/360×π×6²

= 36π/6 cm²

Now, to find the difference in the areas of regions I and III.

The difference in the areas of regions I and III= Area of regions(I+II)-area of regionsII+III

= 64π/6-36π/6

= (28×22/6×7)

= 443 cm²

Hence, the required answer is 443 cm².

6. Shown below is the trophy shield Rishika received on winning an interschool tennis tournament.

(Note: The figure is not to scale.)

The trophy is made of a glass sector DOC supported by identical wooden right triangles, △DAO and △COB. AO = 7 cm and AO:DA = 1:√3.

Find the area of:

i) the glass sector correct to 2 decimal places.

ii) the wooden triangles correct to 2 decimal places.

Show your steps.

(Note: Take п as 22/7 and √3 as 1.73.)

Soln. 

In ∆DOA,

Let ∠DOA=θ⇒tan θ=DA/OA=√3/1

⇒tan θ=√3⇒θ=60°

∴∠DOA=∠COB=60°

∠DOA+∠COB+∠DOC=180°Angles in a straight line⇒∠DOC=180°-60°-60°⇒∠DOC=60°

We know that cos 60°=AO/DO

⇒1/2=7/DO⇒DO=14 cm

Given, AO=7 cm, AO:DA=1:3

⇒7/DA=1/√3⇒DA=7√3cm

Area of the glass sector =θ360×πr²

⇒60°/360×π×14²⇒16×π×196⇒102.67 cm²

Area of the wooden triangle =12×b×h

Area of the wooden triangles=12×7×73×2=84.77 cm²

Important Concepts to Understand in Areas Related to Circles

Before starting areas related to circles, class 10 important questions, students need to understand a few important concepts that form the base of these questions. These concepts encompass the area and circumference of circles, the area of sectors and segments, and the perimeter of these sections.

Area of a Circle

The formula for the area of a circle is one of the first and most basic formulas you'll learn.

Area= πr²

Here, r represents the radius of the circle and π is approximately 3.14159. This formula helps calculate the space contained within the circle’s boundary. In exams, students will often encounter problems where they need to find the area of circular objects like wheels, plates, or even park designs.

Circumference of a Circle

Next up is the circumference, which is essentially the perimeter or boundary of the circle. The formula is:

Circumference= 2πr

This formula gives the length around the circle. Understanding the circumference is just as important as the area because many real-life problems involve the distance around a circular object.

Area of a Sector of a Circle

When you cut out a slice of a circle, like a slice of pizza or a piece of pie, what you have is called a sector. The formula for the area of a sector is:

Area of Sector= ( θ/360°) x πr²

In this formula, θ represents the angle of the sector, and r is the radius. This formula helps calculate the area of just a portion of the circle.

Area of a Segment of a Circle

Now we move on to segments, which are a bit more complex. A segment is the area between a chord (a straight line cutting through the circle) and the arc (the curved line above the chord). The formula for calculating a segment's area is as follows:

Area of Segment= ( θ/360°) x πr² - ½ r²sin(θ)

This formula subtracts the area of the triangle formed by the chord and the radii from the area of the sector.

Perimeter of a Sector

Along with area calculations, students are often asked to calculate the perimeter of a sector. The formula for this is:

Perimeter of Segment= 2r + ( θ/360°) x 2πr

This formula adds the length of the arc (the curved part) to the two straight radii that form the sector.

Why Mastering This Chapter Is Essential for Class 10 Students

• Geometry, and particularly circle-related calculations, are fundamental for higher mathematics. This chapter sets the stage for more advanced topics that students will encounter in future studies.
• In the Class 10 board exams, questions from this chapter and Areas Related to Circles Class 10 Extra Questions with Answers are commonly asked. These can range from direct calculations of areas and perimeters to complex word problems involving sectors and segments. Mastery of this topic can significantly boost your overall math score.
• Learning how to calculate the areas of circles and their parts enhances your problem-solving skills. These skills extend beyond the classroom and are useful in everyday tasks, as well as in technical fields like engineering and architecture.
• Many complex shapes in geometry can be broken down into circles or parts of circles. Being able to calculate the area and perimeter of these basic components makes it easier to tackle composite shapes.

The chapter on "Areas Related to Circles" is an essential part of the Class 10 math syllabus, providing students with both theoretical knowledge and practical problem-solving skills. From calculating the area of a circle to determining the perimeter of a sector, mastering these concepts will help you excel in your exams and develop a solid foundation for future studies in mathematics and related fields.

By focusing on understanding the underlying principles and practicing a variety of problems, students will not only do well in their exams but also gain valuable skills they can apply in real-world situations. So dive deep into this chapter, work through the concepts, and circle back with confidence when it’s time for exams!