Online Self Test
Learn & Practice Quizzes & Challenges Competitive Exams
 

Areas Related to Circles – CBSE NCERT Study Resources

Previous ChapterNext Chapter
Previous Chapter Next Chapter

Study Materials

10th

10th - Mathematics

Areas Related to Circles

Jump to Question Sets

Overview of the Chapter: Areas Related to Circles

This chapter explores the concepts related to the areas of circles and their applications. Students will learn about the perimeter and area of a circle, areas of sectors and segments, and combinations of plane figures involving circles. The chapter builds upon the foundational knowledge of circles from previous grades and extends it to solve real-world problems.

Circle: A circle is the set of all points in a plane that are at a fixed distance from a given point called the center.

Key Concepts

  • Perimeter and Area of a Circle
  • Areas of Sector and Segment of a Circle
  • Areas of Combinations of Plane Figures

Perimeter and Area of a Circle

The perimeter (circumference) of a circle is given by the formula: C = 2πr, where r is the radius of the circle. The area of a circle is given by: A = πr².

Circumference: The distance around the boundary of a circle is called its circumference.

Areas of Sector and Segment of a Circle

A sector of a circle is the region bounded by two radii and the corresponding arc. The area of a sector with central angle θ (in degrees) is given by: (θ/360) × πr².

A segment of a circle is the region bounded by a chord and the corresponding arc. The area of a segment can be calculated by subtracting the area of the corresponding triangle from the area of the sector.

Sector: A part of a circle enclosed by two radii and an arc.

Areas of Combinations of Plane Figures

This section involves calculating areas of complex shapes formed by combining circles with other plane figures such as squares, rectangles, and triangles. The key is to break down the problem into simpler shapes whose areas can be calculated individually and then combined or subtracted as needed.

Summary

The chapter "Areas Related to Circles" equips students with the knowledge to calculate the areas and perimeters of circles, sectors, and segments. It also enhances problem-solving skills by applying these concepts to combinations of plane figures, preparing students for more advanced topics in geometry.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
Find the length of an arc subtending 90° in a circle of radius 4 cm.
Answer:
6.28 cm
Question 2:
What is the area of a quadrant of a circle with radius 10 cm?
Answer:
78.5 cm²
Question 3:
If the circumference of a circle is 31.4 cm, find its radius.
Answer:
5 cm
Question 4:
Find the area of a ring with inner radius 3 cm and outer radius 5 cm.
Answer:
50.24 cm²
Question 5:
What is the perimeter of a semicircle with radius 7 cm?
Answer:
36 cm
Question 6:
Calculate the area of a circle inscribed in a square of side 14 cm.
Answer:
154 cm²
Question 7:
Find the radius of a circle whose area is 154 cm².
Answer:
7 cm
Question 8:
What is the angle subtended by an arc of length 11 cm in a circle of radius 7 cm?
Answer:
90°
Question 9:
Find the area of the shaded region if a square of side 4 cm is inscribed in a circle.
Answer:
9.12 cm²
Question 10:
Find the area of a circle with radius 7 cm.
Answer:
154 cm²
Question 11:
What is the circumference of a circle with diameter 14 cm?
Answer:
44 cm
Question 12:
Calculate the area of a sector with angle 60° and radius 6 cm.
Answer:
18.85 cm²
Question 13:
Find the area of a circle whose radius is 7 cm.
Answer:

The area of a circle is calculated using the formula πr².
Given radius r = 7 cm.
Area = π × (7)² = 22/7 × 49 = 154 cm².

Question 14:
What is the circumference of a circle with diameter 14 cm?
Answer:

The circumference of a circle is given by πd.
Given diameter d = 14 cm.
Circumference = 22/7 × 14 = 44 cm.

Question 15:
Calculate the area of a sector of a circle with radius 6 cm and central angle 60°.
Answer:

The area of a sector is given by (θ/360°) × πr².
Given r = 6 cm and θ = 60°.
Area = (60/360) × 22/7 × 36 = 18.857 cm² (approx).

Question 16:
Find the length of an arc of a circle with radius 10 cm subtending a central angle of 45°.
Answer:

The length of an arc is calculated using (θ/360°) × 2πr.
Given r = 10 cm and θ = 45°.
Arc length = (45/360) × 2 × 22/7 × 10 = 7.857 cm (approx).

Question 17:
What is the area of the largest triangle that can be inscribed in a semicircle of radius 5 cm?
Answer:

The largest triangle inscribed in a semicircle is a right-angled triangle with the diameter as the hypotenuse.
Base = diameter = 10 cm.
Height = radius = 5 cm.
Area = ½ × base × height = ½ × 10 × 5 = 25 cm².

Question 18:
A circular park has a radius of 21 m. Find the cost of fencing it at ₹15 per meter.
Answer:

First, find the circumference of the park.
Circumference = 2πr = 2 × 22/7 × 21 = 132 m.
Cost of fencing = 132 × 15 = ₹1980.

Question 19:
The area of a circle is 154 cm². Find its diameter.
Answer:

Given area = πr² = 154 cm².
So, r² = 154 × 7/22 = 49.
Radius r = 7 cm.
Diameter = 2r = 14 cm.

Question 20:
A wire is bent into the shape of a square of side 22 cm. If the same wire is bent into a circle, find its radius.
Answer:

Perimeter of square = 4 × side = 4 × 22 = 88 cm.
Circumference of circle = 88 cm.
So, 2πr = 88 → r = 88 × 7/44 = 14 cm.

Question 21:
Find the area of a quadrant of a circle whose circumference is 44 cm.
Answer:

First, find the radius using circumference.
2πr = 44 → r = 44 × 7/44 = 7 cm.
Area of quadrant = ¼ × πr² = ¼ × 22/7 × 49 = 38.5 cm².

Question 22:
The difference between the circumference and radius of a circle is 37 cm. Find its radius.
Answer:

Given: 2πr - r = 37.
Factor out r: r(2π - 1) = 37.
So, r = 37 / (44/7 - 1) = 37 / (37/7) = 7 cm.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
Find the area of a sector of a circle with radius 7 cm and central angle 60°.
Answer:

Area of sector = (θ/360°) × πr²
Given: θ = 60°, r = 7 cm
= (60/360) × (22/7) × 7 × 7
= (1/6) × 22 × 7
= 25.67 cm² (approx)

Question 2:
A horse is tied to a pole with a rope of length 14 m. Calculate the area it can graze.
Answer:

The grazing area is a circle with radius = rope length (14 m).
Area = πr²
= (22/7) × 14 × 14
= 22 × 2 × 14
= 616 m²

Question 3:
The circumference of a circle is 44 cm. Find its area.
Answer:

First, find radius (r):
Circumference = 2πr = 44 cm
⇒ r = 44 / (2 × 22/7) = 7 cm
Now, area = πr²
= (22/7) × 7 × 7
= 154 cm²

Question 4:
A circular park has a path of uniform width 2 m around it. If the radius of the park is 21 m, find the area of the path.
Answer:

Outer radius (R) = 21 + 2 = 23 m
Area of path = π(R² - r²)
= (22/7) × (23² - 21²)
= (22/7) × (529 - 441)
= (22/7) × 88
= 276.57 m² (approx)

Question 5:
The minute hand of a clock is 10.5 cm long. Calculate the area it sweeps in 20 minutes.
Answer:

In 20 minutes, the minute hand sweeps a sector with θ = (20/60) × 360° = 120°.
Area = (θ/360°) × πr²
= (120/360) × (22/7) × 10.5 × 10.5
= 115.5 cm²

Question 6:
Find the perimeter of a semicircle with diameter 14 cm.
Answer:

Perimeter = Half circumference + Diameter
= (πr) + d
= (22/7 × 7) + 14
= 22 + 14
= 36 cm

Question 7:
The area of a circle is 154 cm². Find its circumference.
Answer:

First, find radius (r):
Area = πr² = 154
⇒ r² = 154 × (7/22) = 49
⇒ r = 7 cm
Circumference = 2πr
= 2 × (22/7) × 7
= 44 cm

Question 8:
Two circles have radii 5 cm and 3 cm. Find the radius of a circle whose area is the sum of their areas.
Answer:

Combined area = π(5)² + π(3)²
= 25π + 9π = 34π
New radius (R): πR² = 34π
⇒ R² = 34
⇒ R = √34 ≈ 5.83 cm

Question 9:
A square is inscribed in a circle of radius 7 cm. Find the area of the square.
Answer:

Diagonal of square = Diameter of circle = 14 cm
Let side of square = s
Diagonal = s√2 = 14
⇒ s = 14/√2 = 7√2 cm
Area = s² = (7√2)² = 98 cm²

Question 10:
The difference between circumference and radius of a circle is 37 cm. Find its radius.
Answer:

Given: 2πr - r = 37
⇒ r(2π - 1) = 37
⇒ r(44/7 - 1) = 37
⇒ r(37/7) = 37
⇒ r = 7 cm

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
A circular park of radius 20 m has a path of width 5 m around it. Find the area of the path.
Answer:

To find the area of the path, we subtract the area of the inner circle (park) from the area of the outer circle (park + path).

Radius of inner circle (r) = 20 m
Width of path = 5 m
Radius of outer circle (R) = r + width = 20 + 5 = 25 m

Area of path = πR2 - πr2
= π(252 - 202)
= π(625 - 400)
= 225π m2 (or ≈ 706.5 m2 if π = 3.14)

Question 2:
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment.
Answer:

For a right angle subtended at the center:

Area of sector = θ⁄360° × πr2
= 90°⁄360° × π × 102
= 25π cm2

Area of triangle = 1⁄2 × base × height
= 1⁄2 × 10 × 10 = 50 cm2

Area of minor segment = Area of sector - Area of triangle
= 25π - 50 ≈ 28.5 cm2 (when π ≈ 3.14)

Question 3:
Find the area of a quadrant of a circle whose circumference is 88 cm.
Answer:

First, find radius using C = 2πr:

88 = 2 × 22⁄7 × r
r = 14 cm

Area of full circle = πr2
= 22⁄7 × 14 × 14 = 616 cm2

Area of quadrant = 1⁄4 × area of circle
= 1⁄4 × 616 = 154 cm2

Question 4:
The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of these two circles.
Answer:

Let R be the radius of the new circle.

Area of first circle = π × 82 = 64π cm2
Area of second circle = π × 62 = 36π cm2

Total area = 64π + 36π = 100π cm2

New circle's area: πR2 = 100π
⇒ R2 = 100
⇒ R = 10 cm

Question 5:
A sector of a circle with radius 14 cm has a central angle of 60°. Find the area of the sector.
Answer:

The area of a sector is a fraction of the circle's area based on its central angle.

Step 1: Calculate the area of the full circle.
A = πr² = π × (14)² = 616 cm² (using π = 22/7).

Step 2: Find the fraction of the sector.
Fraction = Central angle / 360° = 60° / 360° = 1/6.

Step 3: Multiply to get the sector area.
Sector area = 616 × (1/6) = 102.67 cm² (approx).

Question 6:
A horse is tied to a pole with a rope of length 21 m. Find the area over which the horse can graze.
Answer:

The grazing area is a circle with the rope as the radius.

Step 1: Identify the radius of the grazing area.
Radius = Length of rope = 21 m.

Step 2: Calculate the area using A = πr².
A = π × (21)² = 1386 m² (using π = 22/7).

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding minor segment. (Use π = 3.14)
Answer:
Introduction

We studied that the area of a segment is the difference between the sector and triangle areas. Here, the chord subtends 90°.


Argument 1
  • Area of sector = (θ/360) × πr² = (90/360) × 3.14 × 10² = 78.5 cm²

Argument 2
  • Area of triangle = ½ × base × height = ½ × 10 × 10 = 50 cm²
  • Area of minor segment = 78.5 - 50 = 28.5 cm²

Conclusion

Thus, the area of the minor segment is 28.5 cm², combining sector and triangle concepts.

Question 2:
The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.
Answer:
Introduction

Our textbook shows that the area of a circle is πr². We need to find a new radius for the combined area.


Argument 1
  • Area of first circle = π × 8² = 64π cm²
  • Area of second circle = π × 6² = 36π cm²

Argument 2
  • Total area = 64π + 36π = 100π cm²
  • New radius: πR² = 100π ⇒ R² = 100 ⇒ R = 10 cm

Conclusion

The required radius is 10 cm, derived by equating areas as per the problem.

Question 3:
A circular park of radius 20 m has a rectangular path of width 2 m running around it. Calculate the area of the path. (Use π = 3.14)
Answer:
Introduction

We studied that the area of a path around a circle involves subtracting the inner area from the outer area. Here, the path is rectangular.


Argument 1
  • Outer radius (R) = 20 m + 2 m = 22 m
  • Area of outer circle = πR² = 3.14 × 22 × 22 = 1519.76 m²

Argument 2
  • Inner radius (r) = 20 m
  • Area of inner circle = πr² = 3.14 × 20 × 20 = 1256 m²
  • Area of path = 1519.76 - 1256 = 263.76 m²

Conclusion

Thus, the area of the rectangular path is 263.76 m², as derived from subtracting the two circular areas.

Question 4:
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment. (Use π = 3.14)
Answer:
Introduction

Our textbook shows that the area of a segment is the difference between the sector and triangle areas. Here, the angle is 90°.


Argument 1
  • Area of sector = (θ/360) × πr² = (90/360) × 3.14 × 10 × 10 = 78.5 cm²
  • Area of triangle = ½ × r × r = ½ × 10 × 10 = 50 cm²

Argument 2
  • Area of minor segment = Sector area - Triangle area = 78.5 - 50 = 28.5 cm²

Conclusion

Hence, the area of the minor segment is 28.5 cm², calculated using sector and triangle formulas.

Question 5:
Derive the formula for the area of a sector of a circle with angle θ and radius r. Show steps.
Answer:
Introduction

Our textbook shows that a sector is part of a circle enclosed by two radii and an arc. We derive its area using the circle's total area.


Argument 1
  • Total area of circle = πr².
  • Total angle in circle = 360°.

Argument 2
  • Area for 1° = πr²/360.
  • Area for θ° = (πr²/360) × θ = (θ/360) × πr².

Conclusion

Thus, area of sector = (θ/360) × πr². This is used in real-life applications like slice sizes.

Question 6:
Derive the formula for the area of a sector of a circle with angle θ and radius r. Explain using NCERT-based logic.
Answer:
Introduction

We know a sector is part of a circle bounded by two radii and an arc. Our textbook shows its area depends on the central angle θ.


Argument 1
  • Full circle area = πr²
  • Angle for full circle = 360°

Argument 2
  • Sector area for θ° = (θ/360) × πr²
  • Example: If θ = 60°, area = (60/360) × πr² = (1/6)πr²

Conclusion

Thus, the formula is (θ/360) × πr². This matches NCERT derivations.

Question 7:
Derive the formula for the area of a sector of a circle with angle θ and radius r. Verify using θ = 60° and r = 7 cm.
Answer:
Introduction
Our textbook shows that a sector is a part of a circle bounded by two radii and an arc.
Argument 1
Full circle area = πr². For θ = 360°, sector area = πr². For θ°, sector area = (θ/360) × πr².
Argument 2
For θ = 60° and r = 7 cm: Area = (60/360) × 3.14 × 7 × 7 = 25.66 cm².
Conclusion
The derived formula is (θ/360) × πr², verified as correct (1m for units).
Question 8:
Derive the formula for the area of a sector of a circle with angle θ. Show steps using our textbook method.
Answer:
Introduction

Our textbook shows that a sector is a part of a circle enclosed by two radii and an arc. Its area depends on the central angle θ.


Argument 1
  • Full circle area = πr²
  • Angle for full circle = 360°

Argument 2
  • Area for θ° = (θ/360) × πr²
  • Example: If θ = 60°, area = (60/360) × πr² = (1/6)πr²

Conclusion

Thus, the formula is (θ/360) × πr². This derivation aligns with NCERT Chapter 12.

Question 9:
A circular park of radius 20 m has a rectangular flower bed at its center. The dimensions of the flower bed are 10 m × 5 m. Find the area of the park excluding the flower bed. (Use π = 3.14)
Answer:

To find the area of the park excluding the flower bed, we need to subtract the area of the rectangular flower bed from the area of the circular park.


Step 1: Calculate the area of the circular park.
Area of circle = πr²
Given radius (r) = 20 m
Area = 3.14 × (20)²
= 3.14 × 400
= 1256 m²

Step 2: Calculate the area of the rectangular flower bed.
Area of rectangle = length × breadth
Given dimensions = 10 m × 5 m
Area = 10 × 5
= 50 m²

Step 3: Subtract the area of the flower bed from the area of the park.
Required area = Area of park - Area of flower bed
= 1256 m² - 50 m²
= 1206 m²

Thus, the area of the park excluding the flower bed is 1206 m².

Question 10:
A wire is bent into the shape of a square of side 22 cm. If the same wire is rebent into a circle, find its radius. (Use π = 22/7)
Answer:

To find the radius of the circle formed from the same wire, we first determine the perimeter of the square (which is equal to the circumference of the circle).


Step 1: Calculate the perimeter of the square.
Perimeter of square = 4 × side
Given side = 22 cm
Perimeter = 4 × 22
= 88 cm

Step 2: Relate the perimeter to the circumference of the circle.
Circumference of circle = Perimeter of square
2πr = 88 cm

Step 3: Solve for the radius (r).
Given π = 22/7
2 × (22/7) × r = 88
(44/7) × r = 88
r = 88 × (7/44)
r = 2 × 7
r = 14 cm

Thus, the radius of the circle formed is 14 cm.

Question 11:
A circular park of radius 20 m is surrounded by a road 5 m wide. Find the area of the road. Use π = 3.14.
Answer:

To find the area of the road surrounding the circular park, we need to calculate the difference between the area of the larger circle (park + road) and the area of the park itself.


Step 1: Calculate the radius of the larger circle (park + road).
Radius of park (r) = 20 m
Width of road = 5 m
Radius of larger circle (R) = 20 m + 5 m = 25 m

Step 2: Calculate the area of the larger circle.
Area = πR² = 3.14 × (25)²
= 3.14 × 625
= 1962.5 m²

Step 3: Calculate the area of the park.
Area = πr² = 3.14 × (20)²
= 3.14 × 400
= 1256 m²

Step 4: Find the area of the road.
Area of road = Area of larger circle - Area of park
= 1962.5 m² - 1256 m²
= 706.5 m²

Thus, the area of the road is 706.5 m².

Question 12:
A chord of a circle of radius 14 cm subtends a right angle at the center. Find the area of the corresponding minor segment. Use π = 22/7.
Answer:

To find the area of the minor segment, we subtract the area of the triangle from the area of the sector.


Step 1: Calculate the area of the sector.
Radius (r) = 14 cm
Angle (θ) = 90° (right angle)
Area of sector = (θ/360°) × πr²
= (90°/360°) × (22/7) × (14)²
= (1/4) × (22/7) × 196
= (1/4) × 22 × 28
= 154 cm²

Step 2: Calculate the area of the triangle.
Since the chord subtends a right angle at the center, the triangle formed is a right-angled isosceles triangle with two sides equal to the radius.
Area of triangle = (1/2) × base × height
= (1/2) × 14 × 14
= 98 cm²

Step 3: Find the area of the minor segment.
Area of minor segment = Area of sector - Area of triangle
= 154 cm² - 98 cm²
= 56 cm²

Thus, the area of the minor segment is 56 cm².

Question 13:
A circular park of radius 20 m is surrounded by a road 7 m wide. Find the area of the road. Use π = 22/7.
Answer:

To find the area of the road surrounding the circular park, we need to calculate the difference between the area of the larger circle (park + road) and the smaller circle (park).


Step 1: Calculate the radius of the larger circle (R).
Given: Radius of park (r) = 20 m, Width of road = 7 m.
R = r + width of road = 20 m + 7 m = 27 m.

Step 2: Calculate the area of the larger circle.
Area = πR² = (22/7) × 27 × 27 = (22/7) × 729 ≈ 2291.14 m².

Step 3: Calculate the area of the smaller circle (park).
Area = πr² = (22/7) × 20 × 20 ≈ 1257.14 m².

Step 4: Find the area of the road.
Area of road = Area of larger circle - Area of smaller circle = 2291.14 m² - 1257.14 m² = 1034 m².

Key Concept: The problem uses the formula for the area of a circle and subtracts the inner area to find the annular region (road).
Question 14:
A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the corresponding minor segment. Use π = 22/7.
Answer:

To find the area of the minor segment, we subtract the area of the triangle from the area of the sector.


Step 1: Calculate the area of the sector.
Given: Radius (r) = 14 cm, Angle (θ) = 90°.
Area of sector = (θ/360) × πr² = (90/360) × (22/7) × 14 × 14 = 154 cm².

Step 2: Calculate the area of the triangle.
Since the chord subtends a right angle, the triangle is right-angled.
Area of triangle = (1/2) × base × height = (1/2) × 14 × 14 = 98 cm².

Step 3: Find the area of the minor segment.
Area of segment = Area of sector - Area of triangle = 154 cm² - 98 cm² = 56 cm².

Key Concept: The problem combines the area of a sector and a triangle to find the segment area. Note: The chord divides the circle into a minor and major segment; here, we focus on the smaller (minor) segment.
Question 15:
A circular park of radius 20 m has a rectangular path of width 2 m running around it on the outside. Calculate the area of the path. Also, find the cost of paving the path with tiles at ₹50 per m². Show all steps.
Answer:

To find the area of the path and the cost of paving it, follow these steps:


Step 1: Calculate the area of the circular park
Radius of the park (r) = 20 m
Area of the park = πr² = π × (20)² = 400π m²

Step 2: Calculate the radius of the park including the path
Width of the path = 2 m
New radius (R) = 20 m + 2 m = 22 m

Step 3: Calculate the area of the park including the path
Area = πR² = π × (22)² = 484π m²

Step 4: Find the area of the path
Area of the path = Area of (park + path) - Area of the park
= 484π - 400π = 84π m²
≈ 84 × 3.14 = 263.76 m²

Step 5: Calculate the cost of paving the path
Cost per m² = ₹50
Total cost = 263.76 × 50 = ₹13,188

Key Points:
  • The path is the difference between the larger and smaller circles.
  • Always verify units and calculations for accuracy.
Question 16:
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment. (Use π = 3.14)
Answer:

To find the area of the minor segment, follow these steps:


Step 1: Calculate the area of the sector
Radius (r) = 10 cm
Angle (θ) = 90° (right angle)
Area of sector = (θ/360°) × πr²
= (90/360) × 3.14 × (10)²
= 0.25 × 3.14 × 100 = 78.5 cm²

Step 2: Calculate the area of the triangle
Since the chord subtends a right angle, the triangle formed is a right-angled isosceles triangle.
Area of triangle = ½ × base × height
= ½ × 10 × 10 = 50 cm²

Step 3: Find the area of the minor segment
Area of segment = Area of sector - Area of triangle
= 78.5 - 50 = 28.5 cm²

Key Points:
  • A chord subtending a right angle forms a right-angled triangle.
  • The minor segment is the difference between the sector and the triangle.
  • Always double-check angle and radius values.
Question 17:
A circular park of radius 20 m is surrounded by a path of width 5 m. Find the area of the path. Also, explain the steps involved in solving such problems.
Answer:

To find the area of the path surrounding the circular park, we follow these steps:


Step 1: Calculate the area of the inner circle (park)
Given, radius of the park (r) = 20 m
Area of the park = πr2 = π × (20)2 = 400π m2

Step 2: Calculate the radius of the outer circle (park + path)
Width of the path = 5 m
Radius of outer circle (R) = 20 m + 5 m = 25 m

Step 3: Calculate the area of the outer circle
Area of outer circle = πR2 = π × (25)2 = 625π m2

Step 4: Find the area of the path
Area of the path = Area of outer circle - Area of inner circle
= 625π - 400π = 225π m2

Key Concept: The path forms a ring-shaped (annulus) region around the park. The area of such a region is always the difference between the areas of the two concentric circles.

Question 18:
A horse is tied to a peg at one corner of a square-shaped grass field of side 15 m by means of a 7 m long rope. Find the area of the part of the field in which the horse can graze. Also, describe the reasoning behind the solution.
Answer:

The horse can graze in a quarter-circle area of the field because it is tied to a corner of a square field. Here's how we solve it:


Step 1: Understand the grazing area
The rope length (r) = 7 m (radius of the quarter-circle)
Since the field is square and the horse is tied to a corner, the grazing area is a sector of a circle with a central angle of 90° (quarter of a full circle).

Step 2: Calculate the area of the full circle
Area of full circle = πr2 = π × (7)2 = 49π m2

Step 3: Find the quarter-circle area (grazing area)
Grazing area = (1/4) × Area of full circle = (1/4) × 49π = 12.25π m2

Reasoning: The horse can only move in a quarter-circle because the rope is fixed at the corner of the square field. If the horse were tied in the middle of the field, the grazing area would be a full circle.


Application: This problem demonstrates how geometric constraints (like the shape of the field and the tie point) affect the area calculation. Always visualize the scenario to determine the correct sector of the circle involved.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
A circular park of radius 20 m has a path of width 5 m around it. Calculate the area of the path. (Use π = 3.14)
Answer:
Problem Interpretation

We need to find the area of the path surrounding the circular park. The path is a ring-shaped region.

Mathematical Modeling
  • Inner radius (park) = 20 m
  • Outer radius (park + path) = 20 m + 5 m = 25 m
Solution

Area of path = π(R² - r²) = 3.14 × (25² - 20²) = 3.14 × (625 - 400) = 706.5 m².

Question 2:
A horse is tied to a pole with a 14 m long rope. Find the area the horse can graze. (Use π = 22/7)
Answer:
Problem Interpretation

The horse can move in a circular area around the pole. The rope acts as the radius.

Mathematical Modeling
  • Radius (r) = 14 m
Solution

Grazing area = πr² = (22/7) × 14 × 14 = 616 m². Our textbook shows similar problems with animals tied to poles.

Question 3:
A circular park of radius 20 m has a path of width 5 m around it. Find the area of the path. [Diagram: Park with concentric circular path]
Answer:
Problem Interpretation

We need to find the area of the path surrounding the circular park. The park has radius 20 m, and the path is 5 m wide.

Mathematical Modeling

Outer radius (R) = Park radius + Path width = 20 m + 5 m = 25 m.

Solution

Area of path = πR² - πr² = π(25² - 20²) = π(625 - 400) = 225π m². Our textbook shows similar problems.

Question 4:
A horse is tied to a pole with a rope of length 14 m. Calculate the area the horse can graze. [Diagram: Circle with radius 14 m]
Answer:
Problem Interpretation

The horse can graze in a circular area with radius equal to the rope length (14 m).

Mathematical Modeling

Grazing area = Area of circle with radius 14 m.

Solution

Area = πr² = π(14)² = 196π m². We studied this formula in the chapter.

Question 5:
A circular park of radius 20 m has a path of width 5 m around it. Find the area of the path. Use π = 3.14.
Answer:
Problem Interpretation

We need to find the area of the path surrounding the circular park. The park has a radius of 20 m, and the path is 5 m wide.


Mathematical Modeling

The area of the path is the difference between the area of the larger circle (park + path) and the park itself.


Solution
  • Radius of larger circle (R) = 20 m + 5 m = 25 m
  • Area of path = πR² - πr² = 3.14 × (25² - 20²) = 3.14 × 225 = 706.5 m²
Question 6:
A horse is tied to a pole with a rope of length 14 m. Calculate the area the horse can graze. Use π = 22/7.
Answer:
Problem Interpretation

We studied that a horse tied to a pole can graze in a circular area. The rope length (14 m) acts as the radius.


Mathematical Modeling

The grazable area is the area of a circle with radius 14 m.


Solution
  • Area = πr² = (22/7) × 14 × 14
  • Area = 22 × 2 × 14 = 616 m²
Question 7:
A circular park of radius 20 m has a path of width 5 m around it. Find the area of the path and explain the steps to solve it.
Answer:
Problem Interpretation

We need to find the area of the path surrounding the circular park. The park has a radius of 20 m, and the path is 5 m wide.

Mathematical Modeling
  • Outer radius (R) = Park radius + Path width = 20 m + 5 m = 25 m
  • Inner radius (r) = 20 m
Solution

Area of path = π(R² - r²) = π(25² - 20²) = π(625 - 400) = 225π m².

Question 8:
A horse is tied to a pole with a rope of length 14 m. Calculate the area the horse can graze and verify using the formula for area of a sector.
Answer:
Problem Interpretation

The horse can graze in a circular area with radius equal to the rope length (14 m).

Mathematical Modeling
  • Radius (r) = 14 m
  • Full angle (θ) = 360°
Solution

Area = πr² = π(14)² = 196π m². Our textbook shows this matches the sector area formula (θ/360) × πr² when θ = 360°.

Question 9:
A circular park of radius 20 m has a flower bed in the shape of a sector with angle 60° at the center. Find the area of the flower bed. Also, calculate the remaining area of the park excluding the flower bed. (Use π = 3.14)
Answer:
Problem Interpretation

We need to find the area of a sector (flower bed) and the remaining area of the circular park.

Mathematical Modeling
  • Total park area = πr²
  • Flower bed area = (θ/360°) × πr²
Solution

Total area = 3.14 × 20² = 1256 m². Flower bed area = (60/360) × 1256 ≈ 209.33 m². Remaining area = 1256 - 209.33 ≈ 1046.67 m².

Question 10:
A horse is tied to a corner of a square field of side 14 m with a rope 7 m long. Find the area the horse can graze. Also, determine the ungrazed area of the field. (Use π = 22/7)
Answer:
Problem Interpretation

We must calculate the grazable sector area (90° as it's a corner) and the ungrazed part of the square field.

Mathematical Modeling
  • Grazing area = (θ/360°) × πr²
  • Ungrazed area = Square area - Sector area
Solution

Grazing area = (90/360) × (22/7) × 7² = 38.5 m². Square area = 14² = 196 m². Ungrazed area = 196 - 38.5 = 157.5 m².

Question 11:
A circular park of radius 20 m has a flower bed in the shape of a sector with a central angle of 60° at its center. Calculate the area of the flower bed. Also, find the area of the remaining part of the park. (Use π = 3.14)
Answer:

Given: Radius of the park (r) = 20 m, Central angle (θ) = 60°

Step 1: Calculate the area of the flower bed (sector).
Area of sector = (θ/360°) × πr²
= (60°/360°) × 3.14 × (20)²
= (1/6) × 3.14 × 400
= 209.33 m² (approx)

Step 2: Calculate the total area of the park.
Area of circle = πr²
= 3.14 × (20)²
= 1256 m²

Step 3: Find the remaining area of the park.
Remaining area = Total area - Sector area
= 1256 - 209.33
= 1046.67 m² (approx)

Final Answer:
Area of flower bed = 209.33 m²
Remaining area of park = 1046.67 m²

Question 12:
A wire is bent into the shape of a square of side 22 cm. If the same wire is rebent into a circle, find the radius of the circle. Compare the areas of both shapes. (Use π = 22/7)
Answer:

Given: Side of square (a) = 22 cm

Step 1: Calculate the perimeter of the square (which equals the circumference of the circle).
Perimeter of square = 4 × a
= 4 × 22
= 88 cm

Step 2: Find the radius of the circle.
Circumference of circle = 2πr = 88 cm
⇒ r = 88 / (2 × 22/7)
⇒ r = (88 × 7) / 44
⇒ r = 14 cm

Step 3: Compare areas.
Area of square = a² = 22² = 484 cm²
Area of circle = πr² = (22/7) × 14 × 14 = 616 cm²

Final Answer:
Radius of circle = 14 cm
Area of square = 484 cm², Area of circle = 616 cm²
The circle has a larger area than the square.

Question 13:
A circular field has a circumference of 176 m. A path of uniform width 3.5 m runs around it on the outside. Calculate the area of the path. (Use π = 22/7)
Answer:

Given: Circumference of inner circle = 176 m, Width of path (w) = 3.5 m

Step 1: Find the radius of the inner circle.
Circumference = 2πr = 176
⇒ r = 176 / (2 × 22/7)
⇒ r = (176 × 7) / 44
⇒ r = 28 m

Step 2: Calculate the radius of the outer circle (including path).
Outer radius (R) = r + w = 28 + 3.5 = 31.5 m

Step 3: Compute the area of the path.
Area of path = πR² - πr²
= (22/7) × (31.5)² - (22/7) × (28)²
= (22/7) × (992.25 - 784)
= (22/7) × 208.25
= 654.5 m²

Final Answer:
Area of the path = 654.5 m²

Question 14:

A circular park of radius 20 m has a rectangular path of width 2 m running around it outside. A gardener wants to plant flowers along the boundary of the path. Calculate the area of the path where flowers will be planted.

Use π = 3.14

Answer:

Given: Radius of circular park (r) = 20 m, Width of path (w) = 2 m


Step 1: Calculate the radius of the outer circle (including the path).
Outer radius (R) = r + w = 20 m + 2 m = 22 m


Step 2: Find the area of the outer circle.
Area of outer circle = πR² = 3.14 × (22)² = 3.14 × 484 = 1519.76 m²


Step 3: Find the area of the inner circle (park).
Area of inner circle = πr² = 3.14 × (20)² = 3.14 × 400 = 1256 m²


Step 4: Calculate the area of the path.
Area of path = Area of outer circle - Area of inner circle = 1519.76 m² - 1256 m² = 263.76 m²


Final Answer: The area where flowers will be planted is 263.76 m².

Question 15:

A circular pizza is divided into 8 equal slices. The diameter of the pizza is 28 cm. Find the area of each slice. Also, explain whether the central angle of each slice is equal.

Use π = 22/7

Answer:

Given: Diameter of pizza = 28 cm, Number of slices = 8


Step 1: Calculate the radius of the pizza.
Radius (r) = Diameter / 2 = 28 cm / 2 = 14 cm


Step 2: Find the total area of the pizza.
Area of pizza = πr² = (22/7) × (14)² = (22/7) × 196 = 616 cm²


Step 3: Calculate the area of one slice.
Area per slice = Total area / Number of slices = 616 cm² / 8 = 77 cm²


Step 4: Check if the central angles are equal.
Since the pizza is divided into equal slices, each slice will have an equal central angle.
Central angle per slice = 360° / 8 = 45°


Final Answer: The area of each slice is 77 cm², and the central angle of each slice is equal (45°).

Question 16:

A circular park of radius 20 m has a rectangular path of width 2 m running around it outside. Calculate the area of the path. Also, find the cost of paving the path with tiles at ₹50 per m².

Answer:

To find the area of the path, we subtract the area of the park from the area of the park including the path.


Step 1: Calculate the radius of the outer circle (park + path).
Outer radius = Park radius + Path width = 20 m + 2 m = 22 m.

Step 2: Find the area of the outer circle.
Area = πr² = π × (22)² = π × 484 ≈ 1521.14 m² (using π ≈ 3.14).

Step 3: Find the area of the park (inner circle).
Area = πr² = π × (20)² = π × 400 ≈ 1256.64 m².

Step 4: Calculate the area of the path.
Path area = Outer area - Inner area ≈ 1521.14 m² - 1256.64 m² = 264.5 m².

Step 5: Compute the cost of paving.
Cost = Path area × Rate = 264.5 m² × ₹50/m² = ₹13,225.

Key Note: Always ensure units are consistent and calculations are double-checked for precision.
Question 17:

A circular flower bed is surrounded by a 2 m wide circular lawn. If the radius of the flower bed is 7 m, find the area of the lawn. Also, determine the ratio of the area of the lawn to the area of the flower bed.

Answer:

The lawn forms a ring around the flower bed. We calculate its area and compare it with the flower bed.


Step 1: Find the outer radius (flower bed + lawn).
Outer radius = Flower bed radius + Lawn width = 7 m + 2 m = 9 m.

Step 2: Calculate the area of the outer circle.
Area = πr² = π × (9)² = π × 81 ≈ 254.57 m² (using π ≈ 3.14).

Step 3: Find the area of the flower bed (inner circle).
Area = πr² = π × (7)² = π × 49 ≈ 153.94 m².

Step 4: Determine the lawn area.
Lawn area = Outer area - Inner area ≈ 254.57 m² - 153.94 m² = 100.63 m².

Step 5: Compute the ratio.
Ratio = Lawn area : Flower bed area ≈ 100.63 : 153.94 ≈ 1 : 1.53 (simplified).

Tip: Ratios should be simplified to their lowest terms for clarity. Always verify calculations for accuracy.
Question 18:
A circular park of radius 20 m has a flower bed in the shape of a sector with a central angle of 60° at its center. The remaining area is covered with grass.

(a) Calculate the area of the flower bed.
(b) Find the area covered by grass.
(c) If the cost of maintaining the grass is ₹5 per square meter, what is the total maintenance cost?
Answer:

(a) Area of the flower bed (sector):
Given: Radius (r) = 20 m, Central angle (θ) = 60°
Area of sector = (θ/360°) × πr²
= (60/360) × 3.14 × (20)²
= (1/6) × 3.14 × 400
= 209.33 m² (approx)

(b) Area covered by grass:
Total area of the park = πr² = 3.14 × (20)² = 1256 m²
Grass area = Total area − Flower bed area
= 1256 − 209.33
= 1046.67 m² (approx)

(c) Total maintenance cost:
Cost = Grass area × Rate per m²
= 1046.67 × 5
= ₹5,233.35 (approx)

Note: Always ensure units are consistent and calculations are double-checked for precision.

Question 19:
A circular metallic disc of radius 14 cm has a circular hole of radius 7 cm cut out from its center.

(a) Find the area of the remaining disc.
(b) If the disc is melted and recast into a wire of uniform thickness, what is the length of the wire? (Assume no wastage)
Answer:

(a) Area of the remaining disc:
Outer radius (R) = 14 cm, Inner radius (r) = 7 cm
Area of remaining disc = πR² − πr²
= π(14² − 7²)
= 3.14 × (196 − 49)
= 3.14 × 147
= 461.58 cm²

(b) Length of the wire:
Volume of remaining disc = Area × Thickness (assumed as 1 cm for simplicity)
= 461.58 cm³
Let length of wire be L cm and radius = 0.7 cm (example).
Volume of wire = π × (0.7)² × L = 461.58
L = 461.58 / (3.14 × 0.49)
≈ 300 cm (3 m)

Key concept: Conservation of area/volume is applied here. The problem assumes uniform thickness for simplicity.

Question 20:
A circular park of radius 20 m has a rectangular pond inside it. The pond measures 16 m in length and 12 m in width. Calculate the area of the park not covered by the pond. Use π = 3.14.
Answer:

To find the area of the park not covered by the pond, we first calculate the area of the circular park and then subtract the area of the rectangular pond.

Step 1: Area of the circular park = πr²
= 3.14 × (20)²
= 3.14 × 400
= 1256 m²

Step 2: Area of the rectangular pond = length × width
= 16 × 12
= 192 m²

Step 3: Uncovered area = Area of park - Area of pond
= 1256 - 192
= 1064 m²

Thus, the area of the park not covered by the pond is 1064 m².

Question 21:
A circular flower bed is surrounded by a path 2 m wide. If the diameter of the flower bed is 10 m, find the area of the path. Use π = 3.14.
Answer:

To find the area of the path, we subtract the area of the flower bed from the area of the flower bed including the path.

Step 1: Radius of the flower bed = diameter / 2
= 10 / 2
= 5 m

Step 2: Radius including the path = 5 + 2 = 7 m

Step 3: Area of the flower bed = πr²
= 3.14 × (5)²
= 3.14 × 25
= 78.5 m²

Step 4: Area including the path = πR²
= 3.14 × (7)²
= 3.14 × 49
= 153.86 m²

Step 5: Area of the path = Area including path - Area of flower bed
= 153.86 - 78.5
= 75.36 m²

Thus, the area of the path is 75.36 m².

Question 22:
A circular flower bed is surrounded by a path 2 m wide. The diameter of the flower bed is 14 m. Find the area of the path. Use π = 22/7.
Answer:

To find the area of the path, we subtract the area of the flower bed from the area of the larger circle (flower bed + path).


Step 1: Find the radius of the flower bed
Diameter = 14 m
Radius (r) = 14/2 = 7 m

Step 2: Find the radius of the larger circle (including path)
Width of path = 2 m
Radius (R) = 7 + 2 = 9 m

Step 3: Calculate the area of the flower bed
Area = πr²
= (22/7) × 7 × 7
= 154 m²

Step 4: Calculate the area of the larger circle
Area = πR²
= (22/7) × 9 × 9
= (22/7) × 81
= 254.57 m² (approx)

Step 5: Subtract to find the path's area
Path area = 254.57 - 154
= 100.57 m²

The area of the path is approximately 100.57 m².

Chat on WhatsApp