Surface Areas and Volumes — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)

Step 1: Cuboid dimensions: Length (l) = 12 cm Breadth (b) = 8 cm Height (h) = 6 cm Step 2: Total Surface Area of Cuboid: Formula: Total Surface Area = 2(lb + bh + hl) Step 3: Calculate each product: lb = 12 × 8 = 96 cm² bh = 8 × 6 = 48 cm² hl = 6 × 12 = 72 cm² Step 4: Sum of products: lb + bh + hl = 96 + 48 + 72 = 216 cm² Step 5: Calculate surface area: Total Surface Area = 2 × 216 = 432 cm² Step 6: Volume of Cuboid: Formula: Volume = l × b × h Volume = 12 × 8 × 6 Volume = 96 × 6 Volume = 576 cm³ Step 7: Verification: The cuboid has: - 2 rectangular faces of dimensions 12 × 8 cm - 2 rectangular faces of dimensions 8 × 6 cm - 2 rectangular faces of dimensions 6 × 12 cm Total surface area = 2(96) + 2(48) + 2(72) = 192 + 96 + 144 = 432 cm² ✓ Answer: Total Surface Area = 432 cm²; Volume = 576 cm³.

Step 1: Cylinder dimensions: Radius (r) = 7 cm Height (h) = 10 cm Step 2: Curved Surface Area (Lateral Surface Area): Formula: Curved Surface Area = 2πrh Curved Surface Area = 2 × π × 7 × 10 Curved Surface Area = 140π cm² Curved Surface Area ≈ 140 × 3.14159 ≈ 439.82 cm² Step 3: Area of Two Circular Bases: Area of one circle = πr² Area of one circle = π × 7² Area of one circle = 49π cm² Area of two circles = 2 × 49π = 98π cm² Area of two circles ≈ 98 × 3.14159 ≈ 307.88 cm² Step 4: Total Surface Area: Total Surface Area = Curved Surface Area + Area of two bases Total Surface Area = 140π + 98π Total Surface Area = 238π cm² Total Surface Area ≈ 238 × 3.14159 ≈ 747.70 cm² Step 5: Volume of Cylinder: Formula: Volume = πr²h Volume = π × 7² × 10 Volume = π × 49 × 10 Volume = 490π cm³ Volume ≈ 490 × 3.14159 ≈ 1539.38 cm³ Answer: Curved Surface Area = 140π cm² ≈ 439.82 cm²; Total Surface Area = 238π cm² ≈ 747.70 cm²; Volume = 490π cm³ ≈ 1539.38 cm³.

Step 1: Cone dimensions: Base radius (r) = 5 cm Height (h) = 12 cm Slant height (l) = ? Step 2: Find slant height using Pythagoras theorem: The slant height, radius, and height form a right triangle. l² = r² + h² l² = 5² + 12² l² = 25 + 144 l² = 169 l = 13 cm Step 3: Curved Surface Area (Lateral Surface Area): Formula: Curved Surface Area = πrl Curved Surface Area = π × 5 × 13 Curved Surface Area = 65π cm² Curved Surface Area ≈ 65 × 3.14159 ≈ 204.20 cm² Step 4: Area of Circular Base: Area of base = πr² Area of base = π × 5² Area of base = 25π cm² Area of base ≈ 25 × 3.14159 ≈ 78.54 cm² Step 5: Total Surface Area: Total Surface Area = Curved Surface Area + Base Area Total Surface Area = 65π + 25π Total Surface Area = 90π cm² Total Surface Area ≈ 90 × 3.14159 ≈ 282.74 cm² Step 6: Volume of Cone: Formula: Volume = (1/3)πr²h Volume = (1/3) × π × 5² × 12 Volume = (1/3) × π × 25 × 12 Volume = (1/3) × 300π Volume = 100π cm³ Volume ≈ 100 × 3.14159 ≈ 314.16 cm³ Answer: Slant height = 13 cm; Curved Surface Area = 65π cm² ≈ 204.20 cm²; Total Surface Area = 90π cm² ≈ 282.74 cm²; Volume = 100π cm³ ≈ 314.16 cm³.

Step 1: Sphere radius: r = 6 cm Step 2: Surface Area of Sphere: Formula: Surface Area = 4πr² Surface Area = 4 × π × 6² Surface Area = 4 × π × 36 Surface Area = 144π cm² Surface Area ≈ 144 × 3.14159 ≈ 452.39 cm² Step 3: Volume of Sphere: Formula: Volume = (4/3)πr³ Volume = (4/3) × π × 6³ Volume = (4/3) × π × 216 Volume = (4/3 × 216) × π Volume = (864/3) × π Volume = 288π cm³ Volume ≈ 288 × 3.14159 ≈ 904.78 cm³ Step 4: Verification: For r = 6: 4πr² = 4π(6)² = 144π ✓ (4/3)πr³ = (4/3)π(6)³ = (4/3)π(216) = 288π ✓ Answer: Surface Area = 144π cm² ≈ 452.39 cm²; Volume = 288π cm³ ≈ 904.78 cm³.

Step 1: Hemisphere radius: r = 8 cm Step 2: Curved Surface Area (curved part only): Formula: Curved Surface Area = 2πr² Curved Surface Area = 2 × π × 8² Curved Surface Area = 2 × π × 64 Curved Surface Area = 128π cm² Curved Surface Area ≈ 128 × 3.14159 ≈ 402.12 cm² Step 3: Area of Circular Base: Area of base = πr² Area of base = π × 8² Area of base = 64π cm² Area of base ≈ 64 × 3.14159 ≈ 201.06 cm² Step 4: Total Surface Area: Total Surface Area = Curved Surface Area + Base Area Total Surface Area = 128π + 64π Total Surface Area = 192π cm² Total Surface Area ≈ 192 × 3.14159 ≈ 603.19 cm² Step 5: Volume of Hemisphere: Formula: Volume = (2/3)πr³ Volume = (2/3) × π × 8³ Volume = (2/3) × π × 512 Volume = (2/3 × 512) × π Volume = (1024/3) × π Volume ≈ 341.33π cm³ Volume ≈ 341.33 × 3.14159 ≈ 1072.33 cm³ Step 6: Relationship check: A full sphere with radius 8 has volume (4/3)π(8)³ = (4/3)π(512) = (2048/3)π cm³ A hemisphere is half: (1/3) × (2048/3)π = (1024/3)π cm³ ✓ Answer: Curved Surface Area = 128π cm² ≈ 402.12 cm²; Total Surface Area = 192π cm² ≈ 603.19 cm²; Volume = (1024/3)π cm³ ≈ 1072.33 cm³.

Step 1: Cube side length: a = 5 cm Step 2: Total Surface Area of Cube: A cube has 6 equal square faces. Formula: Total Surface Area = 6a² Total Surface Area = 6 × 5² Total Surface Area = 6 × 25 Total Surface Area = 150 cm² Step 3: Lateral Surface Area (area of 4 vertical faces): Formula: Lateral Surface Area = 4a² Lateral Surface Area = 4 × 5² Lateral Surface Area = 4 × 25 Lateral Surface Area = 100 cm² Step 4: Verification: Total Surface Area = Lateral Surface Area + 2 × (Base Area) 150 = 100 + 2 × 25 150 = 100 + 50 150 = 150 ✓ Step 5: Volume of Cube: Formula: Volume = a³ Volume = 5³ Volume = 125 cm³ Step 6: Diagonal of the cube: Body diagonal = a√3 = 5√3 cm ≈ 8.66 cm (This is the distance from one corner to the opposite corner through the interior.) Answer: Total Surface Area = 150 cm²; Lateral Surface Area = 100 cm²; Volume = 125 cm³.

Step 1: Cylinder with radius 4 cm and height 9 cm: Volume of cylinder = πr²h Volume of cylinder = π × 4² × 9 Volume of cylinder = π × 16 × 9 Volume of cylinder = 144π cm³ Step 2: Cone with radius 4 cm and height 9 cm: Volume of cone = (1/3)πr²h Volume of cone = (1/3) × π × 4² × 9 Volume of cone = (1/3) × π × 16 × 9 Volume of cone = (1/3) × 144π Volume of cone = 48π cm³ Step 3: Compare the volumes: Ratio = Volume of cylinder / Volume of cone Ratio = 144π / 48π Ratio = 144 / 48 Ratio = 3 Step 4: Relationship: Volume of cylinder = 3 × Volume of cone Or: Volume of cone = (1/3) × Volume of cylinder Step 5: Physical interpretation: If you fill a cone with water and pour it into a cylinder with the same base radius and height, it takes exactly 3 cones to fill the cylinder. Step 6: Numerical values: Volume of cylinder ≈ 144 × 3.14159 ≈ 452.39 cm³ Volume of cone ≈ 48 × 3.14159 ≈ 150.80 cm³ Ratio ≈ 452.39 / 150.80 ≈ 3 ✓ Answer: Volume of cylinder = 144π cm³ ≈ 452.39 cm³; Volume of cone = 48π cm³ ≈ 150.80 cm³. The cylinder's volume is exactly 3 times the cone's volume.

Step 1: Tank dimensions (cuboid): Length (l) = 2 m Breadth (b) = 1.5 m Height (h) = 1 m Step 2: Volume of the tank: Volume = l × b × h Volume = 2 × 1.5 × 1 Volume = 3 m³ Step 3: Convert to liters: Given: 1 m³ = 1000 liters Capacity of tank = 3 × 1000 = 3000 liters Step 4: Additional calculations: Surface area of tank (if we need to know surface area of material used): Total Surface Area = 2(lb + bh + lh) Total Surface Area = 2(2×1.5 + 1.5×1 + 1×2) Total Surface Area = 2(3 + 1.5 + 2) Total Surface Area = 2(6.5) Total Surface Area = 13 m² Step 5: If the tank is filled to 3/4 of its capacity: Water volume = (3/4) × 3 = 2.25 m³ = 2250 liters Answer: The tank can hold 3000 liters of water.