Surface Areas and Volumes | Grade 9th - Mathematics Chapter Summary & NCERT CBSE Study Resources

Study Materials

9th

9th - Mathematics

Surface Areas and Volumes

Overview of the Chapter: Surface Areas and Volumes

This chapter introduces students to the concepts of surface areas and volumes of various three-dimensional shapes. It covers the formulas and methods to calculate these measurements for different geometric figures, including cubes, cuboids, cylinders, cones, and spheres. The chapter also includes practical applications of these concepts in real-life scenarios.

Surface Area: The total area of all the surfaces of a three-dimensional object.

Volume: The amount of space occupied by a three-dimensional object.

Key Topics Covered

Surface Area and Volume of a Cuboid
Surface Area and Volume of a Cube
Surface Area and Volume of a Right Circular Cylinder
Surface Area and Volume of a Right Circular Cone
Surface Area and Volume of a Sphere

Surface Area and Volume of a Cuboid

A cuboid has six rectangular faces. The surface area and volume of a cuboid can be calculated using the following formulas:

Total Surface Area (TSA): 2(lb + bh + hl)
Lateral Surface Area (LSA): 2h(l + b)
Volume: l × b × h

Where l is the length, b is the breadth, and h is the height of the cuboid.

Surface Area and Volume of a Cube

A cube is a special case of a cuboid where all sides are equal. The formulas for surface area and volume of a cube are:

Total Surface Area (TSA): 6a²
Lateral Surface Area (LSA): 4a²
Volume: a³

Where a is the length of each side of the cube.

Surface Area and Volume of a Right Circular Cylinder

A right circular cylinder has two circular bases and a curved surface. The formulas are:

Total Surface Area (TSA): 2πr(r + h)
Curved Surface Area (CSA): 2πrh
Volume: πr²h

Where r is the radius and h is the height of the cylinder.

Surface Area and Volume of a Right Circular Cone

A right circular cone has a circular base and a curved surface. The formulas are:

Total Surface Area (TSA): πr(l + r)
Curved Surface Area (CSA): πrl
Volume: (1/3)πr²h

Where r is the radius, h is the height, and l is the slant height of the cone.

Surface Area and Volume of a Sphere

A sphere is a perfectly round three-dimensional object. The formulas are:

Surface Area: 4πr²
Volume: (4/3)πr³

Where r is the radius of the sphere.

Practical Applications

The concepts of surface area and volume are widely used in everyday life, such as in construction, packaging, and manufacturing. Understanding these concepts helps in solving real-world problems efficiently.

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:

What is the total surface area of a cube with side length 4 cm?

Answer:

96 cm²

Question 2:

Find the volume of a cuboid with dimensions 5 cm × 3 cm × 2 cm.

Answer:

30 cm³

Question 3:

What is the curved surface area of a cylinder with radius 7 cm and height 10 cm?

Answer:

440 cm²

Question 4:

Calculate the volume of a sphere with radius 3 cm.

Answer:

113.14 cm³

Question 5:

Find the lateral surface area of a cone with slant height 5 cm and radius 3 cm.

Answer:

47.14 cm²

Question 6:

What is the total surface area of a hemisphere with radius 7 cm?

Answer:

462 cm²

Question 7:

A cylindrical tank has a capacity of 3080 cm³. If its height is 20 cm, find its radius.

Answer:

7 cm

Question 8:

Find the volume of a cone with radius 6 cm and height 7 cm.

Answer:

264 cm³

Question 9:

What is the surface area of a cuboid with dimensions 2 cm × 3 cm × 4 cm?

Answer:

52 cm²

Question 10:

Calculate the slant height of a cone with height 12 cm and radius 5 cm.

Answer:

13 cm

Question 11:

Find the volume of a cylinder with radius 14 cm and height 10 cm.

Answer:

6160 cm³

Question 12:

What is the curved surface area of a hemisphere with radius 10 cm?

Answer:

628.57 cm²

Question 13:

What is the total surface area of a cube with side length 5 cm?

Answer:

The total surface area of a cube is calculated using the formula: 6 × (side)².
Given side = 5 cm,
Total surface area = 6 × (5)²
= 6 × 25
= 150 cm².

Question 14:

Find the lateral surface area of a cylinder with radius 7 cm and height 10 cm.

Answer:

The lateral surface area of a cylinder is given by: 2πrh.
Given r = 7 cm, h = 10 cm,
Lateral surface area = 2 × (22/7) × 7 × 10
= 2 × 22 × 10
= 440 cm².

Question 15:

What is the curved surface area of a cone with slant height 13 cm and radius 5 cm?

Answer:

The curved surface area of a cone is given by: πrl.
Given r = 5 cm, l = 13 cm,
Curved surface area = (22/7) × 5 × 13
= (22/7) × 65
= 204.29 cm² (approx).

Question 16:

Find the total surface area of a cuboid with dimensions 6 cm × 4 cm × 2 cm.

Answer:

The total surface area of a cuboid is: 2(lb + bh + hl).
Given l = 6 cm, b = 4 cm, h = 2 cm,
Total surface area = 2[(6×4) + (4×2) + (2×6)]
= 2[24 + 8 + 12]
= 2 × 44
= 88 cm².

Question 17:

Determine the volume of a cylinder with radius 14 cm and height 20 cm.

Answer:

The volume of a cylinder is: πr²h.
Given r = 14 cm, h = 20 cm,
Volume = (22/7) × 14² × 20
= (22/7) × 196 × 20
= 12320 cm³.

Question 18:

What is the lateral surface area of a cube with edge length 8 cm?

Answer:

The lateral surface area of a cube is: 4 × (side)².
Given side = 8 cm,
Lateral surface area = 4 × (8)²
= 4 × 64
= 256 cm².

Question 19:

Calculate the total surface area of a cone with radius 6 cm and slant height 10 cm.

Answer:

The total surface area of a cone is: πr(r + l).
Given r = 6 cm, l = 10 cm,
Total surface area = (22/7) × 6 × (6 + 10)
= (22/7) × 6 × 16
= 301.71 cm² (approx).

Question 20:

Find the volume of a cuboid with dimensions 10 cm × 5 cm × 3 cm.

Answer:

The volume of a cuboid is: l × b × h.
Given l = 10 cm, b = 5 cm, h = 3 cm,
Volume = 10 × 5 × 3
= 150 cm³.

Question 21:

What is the curved surface area of a hemisphere with radius 7 cm?

Answer:

The curved surface area of a hemisphere is: 2πr².
Given r = 7 cm,
Curved surface area = 2 × (22/7) × 7²
= 2 × (22/7) × 49
= 308 cm².

Question 22:

What is the total surface area of a cube with edge length 5 cm?

Answer:

The total surface area of a cube is calculated using the formula 6a², where a is the edge length.
Given a = 5 cm,
Total surface area = 6 × (5 cm)²
= 6 × 25 cm²
= 150 cm².

Question 23:

Find the lateral surface area of a cylinder with radius 7 cm and height 10 cm.

Answer:

The lateral surface area of a cylinder is given by 2πrh.
Given r = 7 cm, h = 10 cm,
Lateral surface area = 2 × (22/7) × 7 cm × 10 cm
= 2 × 22 × 10 cm²
= 440 cm².

Question 24:

Calculate the volume of a sphere with radius 3 cm.

Answer:

The volume of a sphere is calculated using (4/3)πr³.
Given r = 3 cm,
Volume = (4/3) × (22/7) × (3 cm)³
= (4/3) × (22/7) × 27 cm³
= 113.14 cm³ (approx).

Question 25:

What is the curved surface area of a cone with radius 6 cm and slant height 10 cm?

Answer:

The curved surface area of a cone is given by πrl.
Given r = 6 cm, l = 10 cm,
Curved surface area = (22/7) × 6 cm × 10 cm
= (22/7) × 60 cm²
= 188.57 cm² (approx).

Question 26:

Find the total surface area of a cuboid with dimensions 8 cm × 5 cm × 3 cm.

Answer:

The total surface area of a cuboid is calculated as 2(lb + bh + hl).
Given l = 8 cm, b = 5 cm, h = 3 cm,
Total surface area = 2 × [(8×5) + (5×3) + (3×8)] cm²
= 2 × [40 + 15 + 24] cm²
= 2 × 79 cm²
= 158 cm².

Question 27:

Determine the volume of a cylinder with radius 4 cm and height 12 cm.

Answer:

The volume of a cylinder is given by πr²h.
Given r = 4 cm, h = 12 cm,
Volume = (22/7) × (4 cm)² × 12 cm
= (22/7) × 16 cm² × 12 cm
= 603.43 cm³ (approx).

Question 28:

What is the lateral surface area of a cube with edge length 7 cm?

Answer:

The lateral surface area of a cube is calculated as 4a².
Given a = 7 cm,
Lateral surface area = 4 × (7 cm)²
= 4 × 49 cm²
= 196 cm².

Question 29:

Find the total surface area of a hemisphere with radius 5 cm.

Answer:

The total surface area of a hemisphere is 3πr².
Given r = 5 cm,
Total surface area = 3 × (22/7) × (5 cm)²
= 3 × (22/7) × 25 cm²
= 235.71 cm² (approx).

Question 30:

Calculate the volume of a cone with radius 6 cm and height 8 cm.

Answer:

The volume of a cone is given by (1/3)πr²h.
Given r = 6 cm, h = 8 cm,
Volume = (1/3) × (22/7) × (6 cm)² × 8 cm
= (1/3) × (22/7) × 36 cm² × 8 cm
= 301.71 cm³ (approx).

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 total surface area of a cube with side length 5 cm.

Answer:

The total surface area of a cube is calculated using the formula: 6 × (side)².
Given side = 5 cm,
Total surface area = 6 × (5)²
= 6 × 25
= 150 cm².

Question 2:

Calculate the lateral surface area of a cylinder with radius 7 cm and height 10 cm.

Answer:

The lateral surface area of a cylinder is given by: 2πrh.
Given radius (r) = 7 cm, height (h) = 10 cm,
Lateral surface area = 2 × (22/7) × 7 × 10
= 2 × 22 × 10
= 440 cm².

Question 3:

A cone has a slant height of 13 cm and a base radius of 5 cm. Find its curved surface area.

Answer:

The curved surface area of a cone is calculated using: πrl.
Given radius (r) = 5 cm, slant height (l) = 13 cm,
Curved surface area = (22/7) × 5 × 13
= (22/7) × 65
= 204.29 cm² (approx).

Question 4:

Determine the volume of a sphere with radius 3.5 cm.

Answer:

The volume of a sphere is given by: (4/3)πr³.
Given radius (r) = 3.5 cm,
Volume = (4/3) × (22/7) × (3.5)³
= (4/3) × (22/7) × 42.875
= 179.67 cm³ (approx).

Question 5:

A hemispherical bowl has a radius of 14 cm. Calculate its curved surface area.

Answer:

The curved surface area of a hemisphere is: 2πr².
Given radius (r) = 14 cm,
Curved surface area = 2 × (22/7) × (14)²
= 2 × (22/7) × 196
= 1232 cm².

Question 6:

Find the volume of a cylinder with radius 10 cm and height 21 cm.

Answer:

The volume of a cylinder is calculated using: πr²h.
Given radius (r) = 10 cm, height (h) = 21 cm,
Volume = (22/7) × (10)² × 21
= (22/7) × 100 × 21
= 6600 cm³.

Question 7:

A cone has a base radius of 6 cm and height 8 cm. Find its volume.

Answer:

The volume of a cone is given by: (1/3)πr²h.
Given radius (r) = 6 cm, height (h) = 8 cm,
Volume = (1/3) × (22/7) × (6)² × 8
= (1/3) × (22/7) × 36 × 8
= 301.71 cm³ (approx).

Question 8:

Calculate the lateral surface area of a cube with side length 4 cm.

Answer:

The lateral surface area of a cube is: 4 × (side)².
Given side = 4 cm,
Lateral surface area = 4 × (4)²
= 4 × 16
= 64 cm².

Question 9:

A sphere has a surface area of 154 cm². Find its radius.

Answer:

The surface area of a sphere is: 4πr².
Given surface area = 154 cm²,
4 × (22/7) × r² = 154
r² = (154 × 7) / (4 × 22)
r² = 12.25
r = √12.25
= 3.5 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 cylindrical tank has a diameter of 14 m and height of 5 m. Calculate its lateral surface area.

Answer:

The lateral surface area of a cylinder is given by the formula: 2πrh.
Given, diameter = 14 m, so radius (r) = 7 m.
Height (h) = 5 m.

Step 1: Substitute values into the formula.
Lateral Surface Area = 2 × π × 7 × 5

Step 2: Simplify the calculation.
= 2 × 22/7 × 7 × 5
= 2 × 22 × 5
= 220 m².

Thus, the lateral surface area is 220 m².

Question 2:

A cone has a slant height of 10 cm and a base radius of 6 cm. Find its total surface area.

Answer:

The total surface area of a cone is given by: πr(l + r).
Given, radius (r) = 6 cm, slant height (l) = 10 cm.

Step 1: Substitute values into the formula.
Total Surface Area = π × 6 × (10 + 6)

Step 2: Simplify the calculation.
= 22/7 × 6 × 16
= 22/7 × 96
= 301.71 cm² (approx).

Thus, the total surface area is 301.71 cm².

Question 3:

A cuboidal box has dimensions 12 cm × 8 cm × 6 cm. Calculate its volume.

Answer:

The volume of a cuboid is given by: length × breadth × height.
Given, dimensions = 12 cm × 8 cm × 6 cm.

Step 1: Multiply the dimensions.
Volume = 12 × 8 × 6

Step 2: Simplify the calculation.
= 12 × 48
= 576 cm³.

Thus, the volume of the cuboidal box is 576 cm³.

Question 4:

A sphere has a radius of 7 cm. Find its surface area.

Answer:

The surface area of a sphere is given by: 4πr².
Given, radius (r) = 7 cm.

Step 1: Substitute values into the formula.
Surface Area = 4 × π × 7²

Step 2: Simplify the calculation.
= 4 × 22/7 × 49
= 4 × 22 × 7
= 616 cm².

Thus, the surface area of the sphere is 616 cm².

Question 5:

A hemispherical bowl has a radius of 3.5 cm. Calculate its curved surface area.

Answer:

The curved surface area of a hemisphere is given by: 2πr².
Given, radius (r) = 3.5 cm.

Step 1: Substitute values into the formula.
Curved Surface Area = 2 × π × (3.5)²

Step 2: Simplify the calculation.
= 2 × 22/7 × 12.25
= 2 × 22 × 1.75
= 77 cm².

Thus, the curved surface area of the hemispherical bowl is 77 cm².

Question 6:

A cylindrical tank has a radius of 7 m and a height of 3 m. Calculate its total surface area.

Answer:

The total surface area of a cylinder is given by the formula:
2πr(h + r), where r is the radius and h is the height.

Given: r = 7 m, h = 3 m.

Step 1: Calculate the curved surface area (CSA) = 2πrh = 2 × (22/7) × 7 × 3 = 132 m².
Step 2: Calculate the area of the two circular bases = 2πr² = 2 × (22/7) × 7 × 7 = 308 m².
Step 3: Total surface area = CSA + base areas = 132 + 308 = 440 m².

Note: Always verify units and ensure π is taken as 22/7 unless specified otherwise.

Question 7:

A cone has a slant height of 10 cm and a base radius of 6 cm. Find its curved surface area.

Answer:

The curved surface area (CSA) of a cone is given by the formula:
πrl, where r is the radius and l is the slant height.

Given: r = 6 cm, l = 10 cm.

Step 1: Substitute the values into the formula.
CSA = (22/7) × 6 × 10 = (22/7) × 60 ≈ 188.57 cm² (rounded to two decimal places).

Tip: Slant height (l) is different from perpendicular height (h). Always use the correct dimension in formulas.

Question 8:

A cuboid has dimensions 5 cm × 4 cm × 3 cm. Determine its lateral surface area.

Answer:

The lateral surface area (LSA) of a cuboid is calculated using the formula:
2h(l + b), where l, b, and h are length, breadth, and height, respectively.

Given: l = 5 cm, b = 4 cm, h = 3 cm.

Step 1: Substitute the values into the formula.
LSA = 2 × 3 × (5 + 4) = 6 × 9 = 54 cm².

Note: LSA excludes the top and bottom faces. For total surface area, include 2(lb + bh + hl).

Question 9:

A sphere has a diameter of 14 cm. Find its surface area.

Answer:

The surface area of a sphere is given by the formula:
4πr², where r is the radius.

Given: Diameter = 14 cm ⇒ r = 7 cm.

Step 1: Substitute the radius into the formula.
Surface area = 4 × (22/7) × 7 × 7 = 4 × 22 × 7 = 616 cm².

Tip: Remember that diameter = 2 × radius. Always double-check the given dimension.

Question 10:

A hemispherical bowl has a radius of 3.5 cm. Calculate its total surface area.

Answer:

The total surface area of a hemisphere includes the curved part and the base. The formula is:
3πr², where r is the radius.

Given: r = 3.5 cm.

Step 1: Substitute the radius into the formula.
Total surface area = 3 × (22/7) × 3.5 × 3.5 = 3 × 22 × 0.5 × 3.5 = 115.5 cm².

Note: For a hollow hemisphere (no base), the formula reduces to 2πr².

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 cylindrical tank has a radius of 7 m and height of 10 m. Calculate its lateral surface area and total surface area. Compare these values with a cuboidal tank of dimensions 14 m × 11 m × 10 m.

Answer:

Introduction

We studied that surface area calculations depend on the shape of the object. Here, we compare a cylinder and a cuboid.

Argument 1

Cylinder: Lateral SA = 2πrh = 2 × 22/7 × 7 × 10 = 440 m². Total SA = 2πr(r + h) = 748 m².

Argument 2

Cuboid: Lateral SA = 2h(l + b) = 500 m². Total SA = 2(lb + bh + hl) = 688 m².

Conclusion

The cylinder has a larger total surface area due to its curved surface, as shown in NCERT Example 11.2.

Question 2:

Derive the formula for the volume of a cone using the volume of a cylinder from our textbook. Explain the relationship between their dimensions.

Answer:

Introduction

Our textbook shows that a cone and a cylinder share a proportional relationship in volume.

Argument 1

Volume of cylinder = πr²h (NCERT Formula 13.4).
Experimentally, a cone fills 1/3 of a cylinder with the same base and height.

Argument 2

Thus, Volume of cone = (1/3)πr²h.
This matches NCERT Example 13.5 for a cone with r = 7 cm, h = 24 cm.

Conclusion

The derivation confirms that a cone’s volume is one-third of its corresponding cylinder.

Question 3:

A hemispherical dome has an inner radius of 14 m. Calculate the cost of whitewashing its curved surface at ₹5 per m². How does this compare to painting a spherical balloon of radius 7 m?

Answer:

Introduction

We learned that hemispheres and spheres have different surface areas. Here, we compute their painting costs.

Argument 1

Hemisphere: Curved SA = 2πr² = 1232 m². Cost = 1232 × 5 = ₹6160.

Argument 2

Sphere: SA = 4πr² = 616 m². Cost = 616 × 5 = ₹3080.

Conclusion

The hemispherical dome costs double due to its larger curved surface, as in NCERT Exercise 13.4.

Question 4:

A cylindrical tank has a radius of 7 m and height 10 m. Calculate its lateral surface area and total surface area. Explain the steps with proper units.

Answer:

Introduction

We studied that a cylinder has two circular bases and a curved surface. The lateral surface area (LSA) excludes the bases, while total surface area (TSA) includes them.

Argument 1

Given: Radius (r) = 7 m, Height (h) = 10 m.
LSA = 2πrh = 2 × (22/7) × 7 × 10 = 440 m².

Argument 2

TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 748 m².
Our textbook shows similar problems in Chapter 11.

Conclusion

Thus, LSA is 440 m² and TSA is 748 m². Units are crucial in real-life applications like painting tanks.

Question 5:

A hemispherical dome has a radius of 14 cm. Find its curved surface area and total surface area. Compare it with NCERT Example 4.

Answer:

Introduction

A hemisphere is half of a sphere. Its curved surface area (CSA) is half of a sphere's, while TSA includes the base.

Argument 1

Given: Radius (r) = 14 cm.
CSA = 2πr² = 2 × (22/7) × 14 × 14 = 1232 cm².

Argument 2

TSA = 3πr² = 3 × (22/7) × 14 × 14 = 1848 cm².
NCERT Example 4 uses similar logic for a hemispherical bowl.

Conclusion

The CSA is 1232 cm² and TSA is 1848 cm². Such calculations help in designing domes.

Question 6:

A cubical box has a side of 5 m. Find its lateral surface area and total surface area. How is this applicable in packaging?

Answer:

Introduction

A cube has six equal square faces. LSA includes only the four lateral faces, while TSA includes all six.

Argument 1

Given: Side (a) = 5 m.
LSA = 4a² = 4 × 5 × 5 = 100 m².

Argument 2

TSA = 6a² = 6 × 5 × 5 = 150 m².
Our textbook shows this in Chapter 11 for gift boxes.

Conclusion

LSA is 100 m² and TSA is 150 m². Packaging industries use these calculations to estimate material costs.

Question 7:

A cylindrical tank has a radius of 7 m and height 10 m. Calculate its lateral surface area and total surface area. Compare these values with a cuboidal tank of dimensions 14 m × 11 m × 10 m.

Answer:

Introduction

We studied that surface area calculations depend on shapes. Here, we compare a cylinder and a cuboid.

Argument 1

Cylinder: Lateral SA = 2πrh = 2 × 22/7 × 7 × 10 = 440 m². Total SA = 2πr(r + h) = 748 m².

Argument 2

Cuboid: Lateral SA = 2h(l + b) = 500 m². Total SA = 2(lb + bh + hl) = 688 m².

Conclusion

The cylinder has greater total surface area, but the cuboid has a larger lateral surface area.

Question 8:

Derive the formula for the volume of a cone using the volume of a cylinder from our NCERT textbook. Explain with an example where radius = 6 cm and height = 10 cm.

Answer:

Introduction

Our textbook shows that a cone’s volume is one-third of a cylinder with the same base and height.

Argument 1

Cylinder volume = πr²h. Cone volume = (1/3)πr²h.

Argument 2

Example: For r = 6 cm, h = 10 cm, volume = (1/3) × 22/7 × 36 × 10 = 377.14 cm³.

Conclusion

The derivation confirms the relationship, and the example demonstrates its application.

Question 9:

A hemispherical bowl has an inner radius of 9 cm. Find its curved surface area and total surface area. How much liquid (in litres) can it hold?

Answer:

Introduction

We learned that a hemisphere is half of a sphere, and its surface area and volume follow specific formulas.

Argument 1

Curved SA = 2πr² = 2 × 22/7 × 81 = 509.14 cm². Total SA = 3πr² = 763.71 cm².

Argument 2

Volume = (2/3)πr³ = 1527.43 cm³ = 1.53 litres (since 1000 cm³ = 1 litre).

Conclusion

The bowl’s surface areas and capacity are calculated using standard formulas.

Question 10:

A cylindrical tank has a radius of 7 m and height 10 m. Calculate its total surface area and volume. Explain the steps with proper units.

Answer:

Introduction

We studied that a cylinder has two circular bases and a curved surface. The total surface area includes both.

Argument 1

Radius (r) = 7 m, Height (h) = 10 m.
Volume = πr²h = (22/7) × 7 × 7 × 10 = 1540 m³.

Argument 2

Total Surface Area = 2πr(h + r) = 2 × (22/7) × 7 × (10 + 7) = 748 m².

Conclusion

Our textbook shows similar problems. The volume is 1540 m³, and the surface area is 748 m².

Question 11:

A hemispherical dome has a radius of 14 cm. Find its curved surface area and total surface area. Compare it with NCERT Example 1.

Answer:

Introduction

A hemisphere is half of a sphere. Its curved surface area is half of a sphere's, and total surface area includes the base.

Argument 1

Radius (r) = 14 cm.
Curved Surface Area = 2πr² = 2 × (22/7) × 14 × 14 = 1232 cm².

Argument 2

Total Surface Area = 3πr² = 3 × (22/7) × 14 × 14 = 1848 cm².
NCERT Example 1 uses r = 7 cm, but the formula is the same.

Conclusion

The curved surface area is 1232 cm², and the total surface area is 1848 cm², matching NCERT methods.

Question 12:

A cuboidal box has dimensions 5 m × 4 m × 3 m. Find its lateral surface area and volume. How is this useful in real life?

Answer:

Introduction

Cuboids are common in daily life, like boxes. Lateral surface area excludes top and bottom.

Argument 1

Dimensions: l = 5 m, b = 4 m, h = 3 m.
Volume = l × b × h = 5 × 4 × 3 = 60 m³.

Argument 2

Lateral Surface Area = 2h(l + b) = 2 × 3 × (5 + 4) = 54 m².
Real-life use: Packaging, storage calculations.

Conclusion

The volume is 60 m³, and lateral area is 54 m². Such calculations help in logistics and design.

Question 13:

A cylindrical tank has a diameter of 14 m and height of 3.5 m. Calculate the total surface area of the tank. (Use π = 22/7)

Answer:

To find the total surface area of the cylindrical tank, we use the formula: Total Surface Area = 2πr(h + r), where r is the radius and h is the height.

Given:
Diameter = 14 m ⇒ Radius (r) = Diameter / 2 = 14 / 2 = 7 m
Height (h) = 3.5 m
π = 22/7

Step 1: Substitute the values into the formula:
Total Surface Area = 2 × (22/7) × 7 × (3.5 + 7)

Step 2: Simplify the equation:
= 2 × (22/7) × 7 × 10.5
= 2 × 22 × 10.5 (since 7 cancels out with denominator 7)
= 44 × 10.5
= 462 m²

Thus, the total surface area of the cylindrical tank is 462 m². Remember, the total surface area includes both the curved surface and the two circular bases.

Question 14:

A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 44 m, find the cost of painting it at the rate of ₹25 per square meter. (Use π = 22/7)

Answer:

To find the cost of painting the hemispherical dome, we first calculate its curved surface area using the formula: Curved Surface Area = 2πr², where r is the radius.

Given:
Circumference of base = 44 m ⇒ 2πr = 44
π = 22/7
Cost per m² = ₹25

Step 1: Find the radius (r) from the circumference:
2 × (22/7) × r = 44
(44/7) × r = 44
r = 44 × (7/44)
r = 7 m

Step 2: Calculate the curved surface area of the hemisphere:
Curved Surface Area = 2 × (22/7) × (7)²
= 2 × (22/7) × 49
= 2 × 22 × 7
= 308 m²

Step 3: Compute the total cost of painting:
Total Cost = Area × Rate = 308 × 25 = ₹7,700

Therefore, the cost of painting the hemispherical dome is ₹7,700. Note that only the curved surface area is considered for painting, not the base.

Question 15:

A cylindrical tank has a diameter of 14 m and a height of 3.5 m. Calculate the cost of painting its curved surface at the rate of ₹20 per m². Also, explain why painting the surface of such tanks is important in real-life applications.

Answer:

To find the cost of painting the curved surface of the cylindrical tank, we first calculate its curved surface area (CSA) using the formula: CSA = 2πrh, where r is the radius and h is the height.

Given:
Diameter = 14 m → Radius (r) = 14/2 = 7 m
Height (h) = 3.5 m

Step 1: Calculate CSA
CSA = 2 × (22/7) × 7 × 3.5
CSA = 2 × 22 × 3.5 = 154 m²

Step 2: Calculate cost
Cost = CSA × Rate = 154 × 20 = ₹3080

Real-life importance: Painting the surface of tanks prevents rusting and corrosion, especially in metal tanks. It also improves durability and maintains hygiene, which is crucial for storing water or other liquids.

Question 16:

A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 44 m, find the cost of painting it at ₹25 per m². Also, discuss the significance of hemispherical structures in architecture.

Answer:

First, we find the radius of the hemispherical dome using the circumference of its base. The formula for the circumference of a circle is C = 2πr.

Given:
Circumference (C) = 44 m

Step 1: Find radius (r)
44 = 2 × (22/7) × r
r = (44 × 7) / (2 × 22) = 7 m

Step 2: Calculate curved surface area (CSA) of hemisphere
CSA = 2πr² = 2 × (22/7) × 7 × 7 = 308 m²

Step 3: Calculate cost
Cost = CSA × Rate = 308 × 25 = ₹7700

Architectural significance: Hemispherical structures are strong and distribute weight evenly, making them resistant to external forces like wind. They also provide large interior spaces without internal supports, which is useful in buildings like planetariums or religious structures.

Question 17:

A cylindrical tank has a diameter of 14 meters and a height of 3 meters. Calculate its total surface area and explain why understanding surface area is important in real-life applications like painting or manufacturing.

Answer:

To find the total surface area of the cylindrical tank, we use the formula:
Total Surface Area = 2πr(h + r), where r is the radius and h is the height.

Given:
Diameter = 14 m → Radius (r) = 7 m
Height (h) = 3 m

Step 1: Calculate the curved surface area (CSA).
CSA = 2πrh = 2 × (22/7) × 7 × 3 = 132 m²

Step 2: Calculate the area of the two circular bases.
Base Area = 2 × πr² = 2 × (22/7) × 7 × 7 = 308 m²

Step 3: Add CSA and Base Area for the total surface area.
Total Surface Area = 132 + 308 = 440 m²

Understanding surface area is crucial in real life because:

Painting: It helps estimate the amount of paint needed to cover surfaces like walls or tanks.
Manufacturing: Companies use it to determine material requirements for packaging or constructing objects.
Cost Efficiency: Accurate calculations prevent wastage of resources and reduce expenses.

Question 18:

A hemispherical dome has an inner radius of 10 meters. Calculate the cost of whitewashing its inner curved surface at the rate of ₹25 per square meter. Also, explain how such calculations are used in architectural planning.

Answer:

The curved surface area (CSA) of a hemisphere is given by:
CSA = 2πr², where r is the radius.

Given:
Radius (r) = 10 m
Rate = ₹25 per m²

Step 1: Calculate the CSA.
CSA = 2 × (22/7) × 10 × 10 ≈ 628.57 m²

Step 2: Compute the cost.
Total Cost = CSA × Rate = 628.57 × 25 ≈ ₹15,714.25

Such calculations are vital in architectural planning because:

Budgeting: Helps estimate material and labor costs accurately.
Design Accuracy: Ensures proportions and dimensions align with structural requirements.
Resource Management: Prevents over/under-ordering of materials like paint or plaster.

For example, domes in monuments or planetariums require precise surface area measurements for maintenance and aesthetic purposes.

Question 19:

A cylindrical tank has a diameter of 14 meters and a height of 3 meters. Calculate its total surface area and explain the steps involved. Also, find how much metal sheet would be required to make the tank (assuming no wastage).

Answer:

To find the total surface area of the cylindrical tank, we use the formula: Total Surface Area = 2πr(h + r), where r is the radius and h is the height.

Given: Diameter = 14 m, so Radius (r) = 14/2 = 7 m
Height (h) = 3 m

Step 1: Calculate the curved surface area (CSA) using CSA = 2πrh
CSA = 2 × (22/7) × 7 × 3 = 132 m²

Step 2: Calculate the area of the two circular bases using Area = 2πr²
Base Area = 2 × (22/7) × 7 × 7 = 308 m²

Step 3: Add CSA and Base Area to get Total Surface Area
Total Surface Area = 132 + 308 = 440 m²

The metal sheet required to make the tank is equal to the total surface area, which is 440 m². This ensures no wastage as the entire sheet is used efficiently.

Question 20:

A conical tent has a base radius of 6 meters and a slant height of 10 meters. Find its curved surface area and the cost of canvas required to make the tent at ₹50 per square meter. Justify your answer with proper reasoning.

Answer:

The curved surface area of a cone is given by the formula: CSA = πrl, where r is the radius and l is the slant height.

Given: Radius (r) = 6 m
Slant Height (l) = 10 m

Step 1: Calculate CSA using the formula
CSA = (22/7) × 6 × 10 = 188.57 m² (approx)

Step 2: Calculate the cost of canvas
Cost per m² = ₹50
Total Cost = CSA × Cost per m² = 188.57 × 50 = ₹9,428.50

Justification: The curved surface area represents the amount of canvas needed to cover the conical tent. Since the cost is given per square meter, multiplying the CSA by the rate gives the total expense. Rounding off to two decimal places ensures precision in financial calculations.

Question 21:

A cylindrical tank has a diameter of 14 m and a height of 3.5 m. Calculate its lateral surface area and total surface area. Also, find the cost of painting its curved surface at the rate of ₹20 per m². (Use π = 22/7)

Answer:

Given:
Diameter of the cylindrical tank = 14 m
Radius (r) = Diameter/2 = 14/2 = 7 m
Height (h) = 3.5 m

Step 1: Calculate the lateral surface area (curved surface area)
Lateral Surface Area = 2πrh
= 2 × (22/7) × 7 × 3.5
= 2 × 22 × 3.5
= 154 m²

Step 2: Calculate the total surface area
Total Surface Area = 2πr(r + h)
= 2 × (22/7) × 7 × (7 + 3.5)
= 44 × 10.5
= 462 m²

Step 3: Find the cost of painting the curved surface
Cost = Lateral Surface Area × Rate
= 154 × 20
= ₹3080

Final Answer:
Lateral Surface Area = 154 m²
Total Surface Area = 462 m²
Cost of Painting = ₹3080

Question 22:

A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 44 m, find the cost of painting it at the rate of ₹25 per m². (Use π = 22/7)

Answer:

Given:
Circumference of the base of the hemispherical dome = 44 m

Step 1: Find the radius (r) of the dome
Circumference = 2πr
44 = 2 × (22/7) × r
r = (44 × 7)/(2 × 22)
r = 7 m

Step 2: Calculate the curved surface area of the hemisphere
Curved Surface Area = 2πr²
= 2 × (22/7) × 7 × 7
= 2 × 22 × 7
= 308 m²

Step 3: Find the cost of painting
Cost = Curved Surface Area × Rate
= 308 × 25
= ₹7700

Final Answer:
Cost of Painting = ₹7700

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 cylindrical tank has a radius of 7 m and a height of 10 m. Calculate its lateral surface area and total surface area. (Use π = 22/7)

Answer:

Problem Interpretation

We need to find the lateral and total surface area of a cylinder using given dimensions.

Mathematical Modeling

Lateral Surface Area (LSA) = 2πrh
Total Surface Area (TSA) = 2πr(r + h)

Solution

Given: r = 7 m, h = 10 m. LSA = 2 × (22/7) × 7 × 10 = 440 m². TSA = 2 × (22/7) × 7 × (7 + 10) = 748 m².

Question 2:

A cuboidal box has dimensions 5 cm × 4 cm × 3 cm. Find its volume and the area of the tin sheet required to make it.

Answer:

Problem Interpretation

We must calculate the volume and surface area of a cuboid using its length, breadth, and height.

Mathematical Modeling

Volume = l × b × h
Total Surface Area = 2(lb + bh + hl)

Solution

Given: l = 5 cm, b = 4 cm, h = 3 cm. Volume = 5 × 4 × 3 = 60 cm³. TSA = 2(5×4 + 4×3 + 3×5) = 94 cm².

Question 3:

A cuboidal box has dimensions 5 cm × 4 cm × 3 cm. Find its volume and the length of the diagonal.

Answer:

Problem Interpretation

We must calculate the volume and space diagonal of a cuboid using its length, breadth, and height.

Mathematical Modeling

Volume = l × b × h
Diagonal = √(l² + b² + h²)

Solution

Given: l = 5 cm, b = 4 cm, h = 3 cm. Volume = 5 × 4 × 3 = 60 cm³. Diagonal = √(5² + 4² + 3²) = √50 ≈ 7.07 cm.

Question 4:

A cuboidal box has dimensions 5 cm × 4 cm × 3 cm. Find its volume and the length of the diagonal.

Answer:

Problem Interpretation

We must calculate the volume and space diagonal of a cuboid using its length, breadth, and height.

Mathematical Modeling

Volume = l × b × h
Diagonal = √(l² + b² + h²)

Solution

Given: l = 5 cm, b = 4 cm, h = 3 cm. Volume = 5 × 4 × 3 = 60 cm³. Diagonal = √(5² + 4² + 3²) = √50 ≈ 7.07 cm.

Question 5:

A conical tent has a base radius of 6 m and a slant height of 10 m. Find the canvas area required and the volume of air inside. (Use π = 3.14)

Answer:

Problem Interpretation

We studied conical tents in our textbook. Here, we calculate the curved surface area (canvas) and volume.

Mathematical Modeling

Curved Surface Area (CSA) = πrl
Volume = (1/3)πr²h

Solution

Given: r = 6 m, l = 10 m. First, find height (h) using Pythagoras theorem: h = √(10² - 6²) = 8 m. CSA = 3.14 × 6 × 10 = 188.4 m². Volume = (1/3) × 3.14 × 6² × 8 = 301.44 m³.

Question 6:

A cylindrical water tank has a radius of 3.5 m and height of 7 m. Calculate its lateral surface area and total surface area. (Use π = 22/7)

Answer:

Problem Interpretation

We need to find the lateral and total surface area of a cylindrical water tank.

Mathematical Modeling

Lateral Surface Area (LSA) = 2πrh
Total Surface Area (TSA) = 2πr(r + h)

Solution

Given: r = 3.5 m, h = 7 m. LSA = 2 × (22/7) × 3.5 × 7 = 154 m². TSA = 2 × (22/7) × 3.5 × (3.5 + 7) = 231 m².

Question 7:

A hemispherical dome has a radius of 14 cm. Find its curved surface area and total surface area. (Use π = 22/7)

Answer:

Problem Interpretation

We studied that a hemisphere has a curved surface and a flat circular base.

Mathematical Modeling

Curved Surface Area (CSA) = 2πr²
Total Surface Area (TSA) = 3πr² (including base)

Solution

Given: r = 14 cm. CSA = 2 × (22/7) × 14 × 14 = 1232 cm². TSA = 3 × (22/7) × 14 × 14 = 1848 cm².

Question 8:

A conical tent has a base radius of 6 m and a slant height of 10 m. Find the curved surface area and the canvas required to make it. (Use π = 3.14)

Answer:

Problem Interpretation

We studied cones in our textbook. Here, we calculate the curved surface area and canvas needed for a tent.

Mathematical Modeling

Curved Surface Area (CSA) = πrl

Solution

Given: r = 6 m, l = 10 m. CSA = 3.14 × 6 × 10 = 188.4 m². Since no base is mentioned, canvas required = CSA = 188.4 m².

Question 9:

A cuboidal box has dimensions 5 cm × 4 cm × 3 cm. Find its volume and the area of the tin sheet required to make it.

Answer:

Problem Interpretation

We must calculate the volume and surface area of a cuboid using its length, breadth, and height.

Mathematical Modeling

Volume = l × b × h
Surface Area = 2(lb + bh + hl)

Solution

Given: l = 5 cm, b = 4 cm, h = 3 cm. Volume = 5 × 4 × 3 = 60 cm³. Surface Area = 2(20 + 12 + 15) = 94 cm². Units are specified.

Question 10:

A cylindrical tank has a diameter of 14 m and height of 5 m. Calculate its lateral surface area and total surface area. (Use π = 22/7)

Answer:

Problem Interpretation

We need to find the lateral and total surface area of a cylinder.

Mathematical Modeling

Radius (r) = Diameter/2 = 7 m
Height (h) = 5 m

Solution

Lateral Surface Area = 2πrh = 2 × (22/7) × 7 × 5 = 220 m²
Total Surface Area = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 5) = 528 m²

Question 11:

A cuboidal box has dimensions 12 cm × 8 cm × 6 cm. Find its volume and the length of the longest rod that can fit inside it.

Answer:

Problem Interpretation

We studied how to calculate volume and space diagonal of a cuboid.

Mathematical Modeling

Length (l) = 12 cm
Breadth (b) = 8 cm
Height (h) = 6 cm

Solution

Volume = l × b × h = 12 × 8 × 6 = 576 cm³
Longest rod (diagonal) = √(l² + b² + h²) = √(144 + 64 + 36) = √244 ≈ 15.62 cm

Question 12:

A conical tent has a base radius of 6 m and a slant height of 10 m. Find the curved surface area and the canvas required to make it. (π = 3.14)

Answer:

Problem Interpretation

We studied cones in NCERT. Here, we calculate the curved surface area (CSA) and canvas needed for a conical tent.

Mathematical Modeling

CSA of cone = πrl

Solution

Given: r = 6 m, l = 10 m. CSA = 3.14 × 6 × 10 = 188.4 m². Since canvas covers CSA, required canvas = 188.4 m².

Question 13:

A conical tent has a base radius of 5 m and a slant height of 13 m. Find its curved surface area and the cost of canvas required at ₹50 per m².

Answer:

Problem Interpretation

We must calculate the curved surface area of a cone and the cost to cover it.

Mathematical Modeling

Curved Surface Area (CSA) = πrl
Cost = CSA × Rate

Solution

Given: r = 5 m, l = 13 m. CSA = (22/7) × 5 × 13 ≈ 204.29 m². Cost = 204.29 × 50 = ₹10,214.50.

Question 14:

A cuboidal box has dimensions 5 cm × 6 cm × 7 cm. Find its volume and the length of the longest rod that can fit inside it.

Answer:

Problem Interpretation

We need to calculate the volume and space diagonal of a cuboid.

Mathematical Modeling

Volume = l × b × h
Space diagonal = √(l² + b² + h²)

Solution

Given: l = 5 cm, b = 6 cm, h = 7 cm. Volume = 5 × 6 × 7 = 210 cm³. Longest rod = √(5² + 6² + 7²) = √110 ≈ 10.49 cm.

Question 15:

A cylindrical tank has a diameter of 14 m and height of 3 m. It is filled with water up to 2 m. The water is to be transferred into smaller cylindrical containers, each with a diameter of 7 cm and height of 10 cm. Calculate the number of small containers needed to empty the tank completely.

Use π = 22/7

Answer:

Step 1: Calculate the volume of water in the large tank.
Radius of tank (R) = Diameter/2 = 14/2 = 7 m
Height of water (H) = 2 m
Volume = πR²H = (22/7) × 7 × 7 × 2 = 308 m³

Step 2: Convert volume to cm³ for consistency (1 m³ = 10⁶ cm³).
Volume = 308 × 10⁶ cm³

Step 3: Calculate the volume of one small container.
Radius (r) = 7/2 = 3.5 cm
Height (h) = 10 cm
Volume = πr²h = (22/7) × 3.5 × 3.5 × 10 = 385 cm³

Step 4: Divide total volume by small container volume.
Number of containers = 308 × 10⁶ / 385 = 800,000

Note: Always ensure units are consistent. Here, converting all measurements to cm³ avoids errors.

Question 16:

A conical tent is to be made with a base radius of 6 m and slant height of 10 m. The width of the tarpaulin used is 3 m. Calculate the minimum length of tarpaulin required, assuming no wastage during stitching.

Use π = 3.14

Answer:

Step 1: Find the curved surface area (CSA) of the conical tent.
CSA = πrl = 3.14 × 6 × 10 = 188.4 m²

Step 2: The tarpaulin acts as a rectangle when unrolled. Its area must cover the CSA.
Area of tarpaulin = Length × Width = L × 3

Step 3: Equate the areas to find length (L).
L × 3 = 188.4
L = 188.4 / 3 = 62.8 m

Practical Tip: In real-life scenarios, extra length (5-10%) is added for overlaps and seams, but here we assume perfect efficiency as per the question.

Question 17:

A cylindrical tank has a diameter of 14 m and height of 3.5 m. It is used to store water for a housing society. Calculate the cost of painting its curved surface at the rate of ₹20 per m². Also, find the total water it can hold in liters. (Use π = 22/7)

Answer:

Step 1: Find the radius of the cylindrical tank.
Given diameter = 14 m
Radius (r) = Diameter / 2 = 14 / 2 = 7 m

Step 2: Calculate the curved surface area (CSA).
CSA of cylinder = 2πrh
= 2 × (22/7) × 7 × 3.5
= 2 × 22 × 3.5 = 154 m²

Step 3: Compute the painting cost.
Cost = CSA × Rate = 154 × 20 = ₹3080

Step 4: Determine the volume of water it can hold.
Volume = πr²h
= (22/7) × 7 × 7 × 3.5
= 22 × 7 × 3.5 = 539 m³
Convert to liters (1 m³ = 1000 L):
539 × 1000 = 5,39,000 L

Note: Painting cost covers only the curved surface, excluding the top/bottom. Volume calculation assumes full capacity.

Question 18:

A conical tent is made of canvas with a base radius of 6 m and slant height of 10 m. Find the area of canvas required and the cost of stitching it at ₹15 per m². Additionally, calculate the volume of air inside the tent. (Use π = 3.14)

Answer:

Step 1: Calculate the canvas area (lateral surface area).
LSA of cone = πrl
= 3.14 × 6 × 10
= 188.4 m²

Step 2: Compute stitching cost.
Cost = LSA × Rate = 188.4 × 15 = ₹2826

Step 3: Find the height (h) of the tent using Pythagoras’ theorem.
h = √(l² - r²) = √(10² - 6²) = √(64) = 8 m

Step 4: Determine the volume of air.
Volume = (1/3)πr²h
= (1/3) × 3.14 × 6 × 6 × 8
= 3.14 × 96 = 301.44 m³

Note: Canvas area excludes the base. Volume represents the space occupied by air inside the tent.

Question 19:

A cylindrical tank has a diameter of 14 m and height of 5 m. It is filled with water up to 3 m. Calculate the lateral surface area of the wet portion of the tank and the volume of water it contains. (Use π = 22/7)

Answer:

To solve this problem, we need to find two things: the lateral surface area of the wet portion and the volume of water in the tank.

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

Step 2: Calculate the lateral surface area of the wet portion
The wet portion is the part of the tank in contact with water, which has a height of 3 m.
Lateral Surface Area = 2πrh
= 2 × (22/7) × 7 × 3
= 2 × 22 × 3 = 132 m²

Step 3: Calculate the volume of water
Volume of water = πr²h
= (22/7) × 7 × 7 × 3
= 22 × 7 × 3 = 462 m³

Thus, the lateral surface area of the wet portion is 132 m², and the volume of water is 462 m³.

Question 20:

A conical tent has a base radius of 6 m and a slant height of 10 m. Find the curved surface area of the tent and the volume of air it can hold. (Use π = 3.14)

Answer:

This problem involves calculating the curved surface area of a cone and its volume.

Step 1: Calculate the curved surface area (CSA)
CSA of a cone = πrl
Given: r = 6 m, l = 10 m
= 3.14 × 6 × 10
= 188.4 m²

Step 2: Find the height (h) of the cone
Using the Pythagorean theorem: l² = r² + h²
10² = 6² + h²
100 = 36 + h²
h² = 64
h = 8 m

Step 3: Calculate the volume of the cone
Volume = (1/3)πr²h
= (1/3) × 3.14 × 6 × 6 × 8
= (1/3) × 3.14 × 36 × 8
= 3.14 × 12 × 8 = 301.44 m³

The curved surface area of the tent is 188.4 m², and the volume of air it can hold is 301.44 m³.

Question 21:

A cylindrical tank has a diameter of 14 m and height of 3.5 m. It is filled with water up to 2 m height. Calculate the volume of water in the tank. Also, find the total surface area of the wet portion of the tank.

Answer:

To find the volume of water in the cylindrical tank, we use the formula for the volume of a cylinder: V = πr²h.
Given: Diameter = 14 m, so radius r = 7 m.
Height of water = 2 m.

Step 1: Calculate volume of water.
V = π × (7)² × 2
V = (22/7) × 49 × 2
V = 22 × 7 × 2 = 308 m³

Step 2: Find the total surface area of the wet portion.
The wet portion includes the base and the curved surface up to the water level.
Area of base = πr² = (22/7) × 49 = 154 m²
Curved surface area = 2πrh = 2 × (22/7) × 7 × 2 = 88 m²
Total wet surface area = 154 + 88 = 242 m²

Question 22:

A conical tent has a base radius of 6 m and a slant height of 10 m. Find the cost of canvas required to make the tent at ₹50 per m². Also, calculate the volume of air inside the tent.

Answer:

To find the cost of canvas, we first calculate the curved surface area (CSA) of the conical tent using CSA = πrl.
Given: Radius r = 6 m, Slant height l = 10 m.

Step 1: Calculate CSA.
CSA = (22/7) × 6 × 10
CSA = (22/7) × 60 ≈ 188.57 m²

Step 2: Find cost of canvas.
Cost = 188.57 × 50 = ₹9,428.50

Step 3: Calculate volume of air inside the tent using V = (1/3)πr²h.
First, find height h using Pythagoras' theorem: h = √(l² - r²).
h = √(10² - 6²) = √(100 - 36) = √64 = 8 m
V = (1/3) × (22/7) × 6² × 8
V = (1/3) × (22/7) × 36 × 8 ≈ 301.71 m³

Question 23:

A cylindrical tank has a diameter of 14 meters and a height of 5 meters. It is filled with water up to 3 meters. Calculate the volume of water in the tank. Also, find the total surface area of the wet portion of the tank.

Answer:

To find the volume of water in the cylindrical tank, we use the formula for the volume of a cylinder: V = πr²h.
Given: Diameter = 14 m, so radius (r) = 14/2 = 7 m.
Height of water (h) = 3 m.

Step 1: Calculate the volume of water.
V = π × (7)² × 3
V = (22/7) × 49 × 3
V = 22 × 7 × 3
V = 462 cubic meters.

Step 2: Find the total surface area of the wet portion.
The wet portion includes the base and the curved surface up to the water level.

Surface area of base = πr² = (22/7) × 7 × 7 = 154 m².
Curved surface area = 2πrh = 2 × (22/7) × 7 × 3 = 132 m².

Total wet surface area = Base area + Curved surface area = 154 + 132 = 286 m².

Question 24:

A conical tent has a base radius of 6 meters and a slant height of 10 meters. Calculate the cost of the canvas required to make the tent at the rate of ₹50 per square meter. Also, find the volume of air inside the tent.

Answer:

To find the cost of canvas, we first calculate the curved surface area (CSA) of the conical tent using the formula: CSA = πrl.
Given: Radius (r) = 6 m, Slant height (l) = 10 m.

Step 1: Calculate CSA.
CSA = (22/7) × 6 × 10
CSA = (22/7) × 60
CSA ≈ 188.57 m² (approx).

Step 2: Calculate cost.
Cost = CSA × Rate = 188.57 × 50 = ₹9428.50 (approx).

Step 3: Find the volume of air inside the tent using the formula: V = (1/3)πr²h.
First, find height (h) using Pythagoras' theorem: h = √(l² - r²) = √(100 - 36) = √64 = 8 m.

Volume = (1/3) × (22/7) × 6 × 6 × 8
V = (1/3) × (22/7) × 288
V ≈ 301.71 m³ (approx).

Surface Areas and Volumes – CBSE NCERT Study Resources

Overview of the Chapter: Surface Areas and Volumes

Key Topics Covered

Surface Area and Volume of a Cuboid

Surface Area and Volume of a Cube

Surface Area and Volume of a Right Circular Cylinder

Surface Area and Volume of a Right Circular Cone

Surface Area and Volume of a Sphere

Practical Applications

All Question Types with Solutions – CBSE Exam Pattern

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

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

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

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

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