Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

This hard level explores advanced concepts including diagonal of cube, melting and recasting problems, and complex multi-step calculations.

Master these concepts for competitive examinations and real-world engineering applications.

1. What is the relationship between edge and diagonal in a cube?

Diagonal = √3 × aa, where a is the edge

Perfect! The space diagonal of a cube = √3 times its edge.

2. Write the formula for total surface area of a cylinder.

TSA = 2πr² + 2πrhπr²h or 2πr(r + h)πr(r + h)

Excellent! Both forms are correct: 2πr² + 2πrh = 2πr(r + h).

3. Find the number of edges in a cuboid.

Number of edges =

Correct! A cuboid has 12 edges (4 length + 4 breadth + 4 height).

4. Find the lateral surface area of a cube whose side is 10 cm.

LSA = 4a² = 4 × ()² = cm²

Great! Lateral surface area = 400 cm².

5. Define volume.

Volume is the amount of space occupied by a 3D objectthe surface area.

Perfect! Volume measures the capacity or space inside a solid.

Drag each concept to its application:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Find the volume of a cube whose surface area is 150 cm².

Surface area = 6a²6a = 150

a² = 150 ÷ =

a = cm

Volume = a³ = ³ = cm³

Excellent! Volume = 125 cm³.

2. The total surface area of a cube is 216 cm². Find its edge.

TSA = 6a² = 216

a² = 216 ÷ =

a = cm

Perfect! Edge = 6 cm.

3. A cylindrical container has radius 3.5 cm and height 20 cm. Find its total surface area.

TSA = 2πr(r + h)

= 2 × × × ( + )

= 2 × 3.14 × 3.5 ×

= cm² (approximately)

Great! Total surface area ≈ 517 cm².

4. A cuboid has a volume of 120 cm³ and base area 20 cm². Find its height.

Volume = Base area × Height

120 = × h

h = 120 ÷ 20 = cm

Excellent! Height = 6 cm.

5. The height of a cylinder is twice its radius. If the total surface area is 616 cm², find the radius and height.

Let radius = r, then height =

TSA = 2πr(r + h) = 2πr(r + 2r) = 2πr × =

6πr² = 616

6 × × r² = 616

× r² = 616

r² = 616 ÷ 18.84 ≈

r ≈ cm (approximately)

Height = 2r ≈ cm

Perfect! Radius ≈ 5.7 cm and Height ≈ 11.4 cm. (Note: Using π = 22/7 gives r = 7 cm, h = 14 cm exactly).

Part A: Section C – Long Answer Questions (4 Marks Each)

1. A metallic cylinder of radius 7 cm and height 14 cm is melted to form small spherical balls of radius 1.4 cm. Find how many such balls are formed.

Volume of cylinder = πr²hπr²

= × ² ×

= 3.14 × 49 × 14 = cm³

Volume of one sphere = (4/3)πr³

= (4/3) × 3.14 × ()³

= (4/3) × 3.14 × ≈ cm³

Number of balls = Volume of cylinder ÷ Volume of one sphere

= 2154.04 ÷ 11.49 ≈ ≈ balls

Excellent! Approximately 187-188 spherical balls are formed.

2. The internal dimensions of a rectangular tank are 2.5 m × 1.8 m × 1.4 m. Find its capacity in litres.

Volume = l × b × h

= × ×

= m³

Converting to litres: 1 m³ = litres

Capacity = 6.3 × 1000 = litres

Perfect! Capacity = 6300 litres.

3. A solid cube of side 15 cm is cut into smaller cubes of side 5 cm. Find how many small cubes are obtained and their total surface area.

Volume of large cube = ³ = cm³

Volume of each small cube = ³ = cm³

Number of small cubes = 3375 ÷ 125 = cubes

Surface area of one small cube = 6 × ² = 6 × = cm²

Total surface area of all 27 cubes = 27 × 150 = cm²

Excellent! 27 small cubes with total surface area = 4050 cm².

4. A cylindrical tin of radius 10 cm and height 21 cm is to be painted. Find the area to be painted and the cost of painting it at ₹8 per 100 cm².

Area to be painted = Total surface area

TSA = 2πr(r + h)

= 2 × × × ( + )

= 2 × 3.14 × 10 ×

= cm²

Cost = (1948.4 ÷ 100) × 8

= × 8 = ₹

Perfect! Area = 1948.4 cm² and Cost ≈ ₹156.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. If edge of cube = 4 cm, volume =

(a) 8 cm³ (b) 16 cm³ (c) 64 cm³ (d) 32 cm³

Correct! Volume = 4³ = 64 cm³.

7. The total surface area of a cube of edge 3 cm is:

(a) 9 cm² (b) 18 cm² (c) 36 cm² (d) 54 cm²

Perfect! TSA = 6 × 3² = 6 × 9 = 54 cm².

8. A cuboid has l = 5 cm, b = 4 cm, h = 3 cm. TSA =

(a) 62 cm² (b) 94 cm² (c) 94 cm² (d) 100 cm²

Excellent! TSA = 2(5×4 + 4×3 + 3×5) = 2(20 + 12 + 15) = 2 × 47 = 94 cm².

9. Volume of cuboid =

(a) l × b × h (b) 2(l + b)h (c) l + b + h (d) None

Perfect! Volume = length × breadth × height.

10. The height of a cylinder whose volume is 3080 cm³ and radius 7 cm is:

(a) 20 cm (b) 10 cm (c) 15 cm (d) 12 cm

Correct! h = V ÷ (πr²) = 3080 ÷ (3.14 × 49) ≈ 20 cm.

🎉 Exceptional Achievement! You've Mastered Advanced Surface Area and Volume!

Here's what you learned:

• Advanced Formulas and Relationships:

  • Cube Diagonal: √3 × edge (space diagonal)
  • Cylinder TSA: 2πr(r + h) = 2πr² + 2πrh
  • Sphere Volume: (4/3)πr³
  • Conversions: 1 m³ = 1,000,000 cm³ = 1000 litres

• Complex Problem Types:

  1. Melting/Recasting Problems:

  • Volume remains constant when melted
  • Volume of original = Total volume of new shapes
  • Example: Cylinder melted into spheres

  2. Cutting/Division Problems:

  • Original volume = Sum of small volumes
  • Surface area increases when cut
  • Example: Large cube cut into smaller cubes

  3. Reverse Calculations:

  • Finding dimensions from surface area/volume
  • Using formula manipulation
  • Example: Find edge from TSA

  4. Capacity Problems:

  • Convert m³ to litres: multiply by 1000
  • Internal dimensions for capacity
  • Example: Tank capacity in litres

• Key Techniques:

  Finding Unknown Dimensions:

  • From TSA: Use 6a² = TSA → a = √(TSA/6)
  • From Volume: Use a³ = V → a = ³√V
  • From cylinder data: Rearrange V = πr²h

  Melting Problems:

  • Step 1: Find volume of original solid
  • Step 2: Find volume of one new solid
  • Step 3: Divide to find number of pieces

  Cutting Problems:

  • Number of cubes = (Large edge ÷ Small edge)³
  • Surface area increases after cutting
  • Each cut creates new faces

• Complete Formula Sheet:

  Cube (edge = a):

  • TSA = 6a²
  • LSA = 4a²
  • Volume = a³
  • Diagonal = √3 × a

  Cuboid (l, b, h):

  • TSA = 2(lb + bh + hl)
  • LSA = 2(l + b)h
  • Volume = l × b × h

  Cylinder (r, h):

  • TSA = 2πr(r + h)
  • CSA = 2πrh
  • Volume = πr²h

  Sphere (r):

  • TSA = 4πr²
  • Volume = (4/3)πr³

• Unit Conversions:

  • 1 m = 100 cm
  • 1 m² = 10,000 cm²
  • 1 m³ = 1,000,000 cm³
  • 1 m³ = 1000 litres
  • 1 litre = 1000 cm³

• Cost Calculation Strategy:

  1. Find total surface area
  2. Convert to required units (m² or cm²)
  3. Multiply by rate per unit area
  4. Add all costs if multiple surfaces

These advanced concepts are crucial for engineering design, manufacturing, and competitive examinations!