Surface Areas and Volumes — Telangana (SSC) Class 9 Mathematics Solutions (Free)

Given: length (l) = 10 cm, breadth (b) = 6 cm, height (h) = 5 cm. Total Surface Area: TSA = 2(lb + bh + hl) TSA = 2(10×6 + 6×5 + 5×10) TSA = 2(60 + 30 + 50) TSA = 2(140) = 280 cm² Volume: V = l × b × h V = 10 × 6 × 5 = 300 cm³

Given: radius (r) = 14 cm, height (h) = 20 cm. Curved Surface Area: CSA = 2πrh CSA = 2 × (22/7) × 14 × 20 = 1,760 cm² Total Surface Area: TSA = 2πr(r + h) TSA = 2 × (22/7) × 14 × (14 + 20) = 2 × (22/7) × 14 × 34 = 2,992 cm² Volume: V = πr²h V = (22/7) × 14 × 14 × 20 = 12,320 cm³

Given: radius (r) = 7 cm, height (h) = 24 cm. Slant height: l = √(r² + h²) l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm Curved Surface Area: CSA = πrl CSA = (22/7) × 7 × 25 = 550 cm² Total Surface Area: TSA = πr(r + l) TSA = (22/7) × 7 × (7 + 25) = 22 × 32 = 704 cm² Volume: V = (1/3)πr²h V = (1/3) × (22/7) × 7 × 7 × 24 = (1/3) × 22 × 7 × 24 = 1,232 cm³

Given: radius (r) = 7 cm. Surface Area: SA = 4πr² SA = 4 × (22/7) × 7 × 7 = 616 cm² Volume: V = (4/3)πr³ V = (4/3) × (22/7) × 7 × 7 × 7 = (4 × 22 × 49)/3 = 4,312/3 ≈ 1,437.33 cm³

Given: radius (r) = 21 cm. Curved Surface Area: CSA = 2πr² CSA = 2 × (22/7) × 21 × 21 = 2,772 cm² Total Surface Area: TSA = CSA + area of the flat circular base = 2πr² + πr² = 3πr² TSA = 3 × (22/7) × 21 × 21 = 4,158 cm² Volume: V = (2/3)πr³ V = (2/3) × (22/7) × 21 × 21 × 21 = 19,404 cm³

Given: side (a) = 7 cm. Total Surface Area: TSA = 6a² (all six square faces) TSA = 6 × 7² = 6 × 49 = 294 cm² Lateral Surface Area: LSA = 4a² (the four side faces only, excluding top and bottom) LSA = 4 × 49 = 196 cm² Volume: V = a³ V = 7³ = 343 cm³

Given: r = 6 cm, h = 14 cm for both solids. Volume of the cylinder: V_cylinder = πr²h = (22/7) × 6 × 6 × 14 = 1,584 cm³ Volume of the cone: V_cone = (1/3)πr²h = (1/3) × 1,584 = 528 cm³ Comparing: V_cylinder / V_cone = 1,584 / 528 = 3 The relationship: a cone with the SAME base radius and height as a cylinder always has exactly ONE-THIRD the cylinder's volume — this follows directly from their formulas, since the cone's formula is simply the cylinder's formula multiplied by the constant factor 1/3. This is why a cone-shaped container needs to be filled and emptied exactly three times to fill a cylinder of the same base and height.

Step 1: Find the volume of the tank in cubic metres. V = length × breadth × height = 3 × 2 × 1.5 = 9 m³ Step 2: Convert cubic metres to litres. V = 9 × 1,000 = 9,000 litres The tank can hold 9,000 litres of water.