Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Find the volume of a sphere with radius 14 cm. cm3

Correct! Volume = 43πr3 = 43π × 2744 = 10976π3 ≈ 11494.4 cm3.

(2) Find the total surface area of a cone with radius 5 cm and slant height 13 cm. cm2

Perfect! TSA = πr(r + l) = π × 5 × (5 + 13) = 90π ≈ 282.6 cm2.

(3) Find the lateral surface area of a cylinder with radius 7 cm and height 24 cm. cm2

Excellent! LSA = 2πrh = 2π × 7 × 24 = 336π ≈ 1055.04 cm2.

(4) A cube has volume 343 cm3. Find its side.

Great! Side = ∛343 = 7 cm.

(5) Find the radius of a sphere whose volume is 523.6 cm3 (use π ≈ 3.14). cm

Correct! Volume = 43πr3 = 523.6, so r3 = 125, therefore r = 5 cm.

Short Answer Questions (2 Marks Each)

Answer each question with detailed calculations

(1) A cube of side 12 cm is melted to form a sphere. Find the radius of the sphere.

Radius: cm

Excellent! Cube volume = 1728 cm3. Sphere volume = 43πr3 = 1728, so r3 = 413.37, therefore r ≈ 6.2 cm.

(2) A cylinder has radius 10 cm and height 21 cm. Find its lateral surface area and total surface area.

LSA: cm2 TSA: cm2

Perfect! LSA = 2πrh = 2π × 10 × 21 = 420π ≈ 1319.47 cm2, TSA = 2πr(r + h) = 2π × 10 × 31 = 620π ≈ 1947.78 cm2.

(3) A cone has radius 6 cm and height 8 cm. Find its slant height, lateral surface area, and total surface area.

Slant height: cm LSA: cm2 TSA: cm2

Correct! Slant height = 36+64 = 10 cm, LSA = πrl = 60π ≈ 188.4 cm2, TSA = πr(r + l) = 96π ≈ 301.44 cm2.

(4) A cuboid has dimensions 14 cm × 10 cm × 8 cm. Find its total surface area, lateral surface area, and volume.

TSA: cm2 LSA: cm2 Volume: cm3

Great! TSA = 2(140 + 80 + 112) = 636 cm2, LSA = 2h(l + b) = 2 × 8 × 24 = 384 cm2, Volume = 1120 cm3.

(5) A spherical solid of radius 7 cm is melted to form 343 small spheres of equal size. Find the radius of each small sphere.

Radius: cm

Perfect! Large sphere volume = 43π × 343 = 1372π3. Small sphere volume = 1372π3 ÷ 343 = 4π3. So r3 = 1, therefore r = 1 cm.

Long Answer Questions (4 Marks Each)

(1) A metallic sphere of radius 12 cm is melted to make a right circular cone of base radius 8 cm. Find the height of the cone.

Height: cm

Correct! Sphere volume = 43π × 1728 = 2304π cm3. Cone volume = 13π × 64 × h = 2304π. So h = 27 cm.

(2) A cube of side 15 cm is melted to form smaller cubes of side 5 cm. Find the number of smaller cubes formed and the total surface area of all the smaller cubes.

Number of cubes: Total surface area: cm2

Perfect! Large cube volume = 3375 cm3, small cube volume = 125 cm3. Number = 27. Each small cube TSA = 150 cm2, total = 27 × 150 = 4050 cm2.

(3) A solid hemisphere of radius 14 cm is melted to form small spheres of radius 2 cm. Find the number of spheres formed.

Number of spheres:

Excellent! Hemisphere volume = 23π × 2744 = 5488π3. Small sphere volume = 32π3. Number = 5488π3 ÷ 32π3 = 343.

(4) A cylindrical tank has radius 7 m and height 10 m. Find the volume in cubic meters and the total surface area in square meters.

Volume: m3 TSA: m2

Great! Volume = πr2h = π × 49 × 10 = 490π ≈ 1539 m3, TSA = 2πr(r + h) = 2π × 7 × 17 = 238π ≈ 747.7m2.

(5) A cone has radius 21 cm and height 28 cm. Find its slant height, lateral surface area, total surface area, and volume.

Slant height: cm LSA: cm2 TSA: cm2 Volume: cm3

Correct! Slant height = 441+784 = 35 cm, LSA = π × 21 × 35 = 735π ≈ 2309.07 cm2, TSA = π × 21 × 56 = 1176π ≈ 3694.51 cm2, Volume = 13 × π × 441 × 28 = 4116π ≈ 12936 cm3.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The volume of a sphere with radius 7 cm is:

(a) 1436 cm3 (b) 1435 cm3 (c) 1434 cm3 (d) 1437 cm3

Correct! Volume = 43πr3 = 43π × 343 = 1372π3 ≈ 1437 cm3.

(2) The lateral surface area of a cylinder with radius 5 cm and height 14 cm is:

(a) 440 cm2 (b) 380π cm2 (c) 140π cm2 (d) 350π cm2

Wait, let me recalculate: LSA = 2πrh = 2π × 5 × 14 = 140π cm2.

(3) The total surface area of a cone with radius 7 cm and slant height 25 cm is:

(a) 224 cm2 (b) 224π cm2 (c) 252π cm2 (d) 225π cm2

Correct! TSA = πr(r + l) = π × 7 × (7 + 25) = 224π cm2.

(4) A cube has volume 1000 cm3. Its side is:

(a) 10 cm (b) 12 cm (c) 8 cm (d) 15 cm

Correct! Side = ∛1000 = 10 cm.

(5) A sphere of radius 14 cm is melted to form a cone of base radius 7 cm. The height of the cone is:

(a) 48 cm (b) 49 cm (c) 50 cm (d) 52 cm

Let me recalculate: Sphere volume = 43π × 2744. Cone volume = 13π × 49 × h. So h = 4×274449 = 224 cm. The closest given option would be around 224 cm, but none match exactly.

(6) A cube of side 9 cm is melted to form a sphere. Radius of sphere is:

(a) 6.2 cm (b) 5.8 cm (c) 6.0 cm (d) 6.5 cm

Correct! Cube volume = 729 cm3. Sphere volume = 43πr3 = 729, so r ≈ 6.2 cm.

(7) The slant height of a cone with radius 15 cm and height 20 cm is:

(a) 25 cm (b) 26 cm (c) 24 cm (d) 23 cm

Correct! Slant height = 152+202 = 225+400 = 625 = 25 cm.

(8) A cylinder has radius 7 cm and height 24 cm. Its volume is:

(a) 3696 cm3 (b) 3600 cm3 (c) 3700 cm3 (d) 3750 cm3

Correct! Volume = πr2h = π × 49 × 24 = 1176π ≈ 3696 cm3.

(9) The total surface area of a cube with side 8 cm is:

(a) 384 cm2 (b) 256 cm2 (c) 320 cm2 (d) 400 cm2

Correct! TSA =cm²8^2=6×64=384cm^2`.

(10) A hemisphere has radius 14 cm. Its curved surface area is:

(a) 1232 cm2 (b) 1231 cm2 (c) 1225 cm2 (d) 1240 cm2

Correct! Curved surface area = 2πr2 = 2π × 196 = 392π ≈ 1232 cm2.

Expert 3D Geometry Challenge

Determine whether these statements are True or False:

Expert 3D Geometry Mastery Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Expert Surface Areas and Volumes" worksheet and learned:

(1) Advanced Volume Conservation: Mastering complex melting and recasting problems with multiple objects

(2) Large Number Calculations: Working confidently with volumes over 10,000 cubic units

(3) Complex Shape Transformations: Converting between spheres, cones, cylinders, and cubes systematically

(4) Multi-step Problem Solving: Breaking down complex problems into manageable calculation steps

(5) Advanced Hemisphere Geometry: Understanding curved surface areas and volume relationships

(6) Expert Material Applications: Solving real-world problems involving metal casting and manufacturing

(7) Cube Root Mastery: Finding dimensions from given volumes using inverse operations

(8) Advanced Slant Height Calculations: Using Pythagorean theorem in complex cone problems

(9) Industrial-scale Calculations: Working with large tanks, spheres, and manufacturing scenarios

(10) Mathematical Precision: Maintaining accuracy through complex multi-step calculations

Exceptional achievement! You have mastered expert-level 3D geometry and can solve complex real-world problems involving surface areas and volumes with professional-level precision!