Surface Areas and Volumes — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Surface Areas and Volumes" — 9 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Surface Areas and Volumes" — 9 important questions with detailed answers for CBSE…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

A solid is formed by combining a cylinder and a hemisphere at one end. The cy…
A cone is placed on top of a cylinder. The cylinder has radius 5 cm and heigh…
Find the volume of a frustum of a cone if the radii of the two circular ends …
A solid is made by combining a hemisphere and a cone. The hemisphere has radi…
Find the volume of a composite solid formed by joining a cylinder (radius 6 c…
A frustum of a cone has slant height 10 cm, and the radii of the two ends are…
+ 3 more questions in the full chapter

Solutions Summary:

Question	Status
A solid is formed by combining a cylinder and a hemispher…	✓ Solved
A cone is placed on top of a cylinder. The cylinder has r…	✓ Solved
Find the volume of a frustum of a cone if the radii of th…	✓ Solved
A solid is made by combining a hemisphere and a cone. The…	✓ Solved
Find the volume of a composite solid formed by joining a …	✓ Solved
A frustum of a cone has slant height 10 cm, and the radii…	✓ Solved

Showing 6 of 9 questions

Q1: A solid is formed by combining a cylinder and a hemisphere at one end. The cylinder has radius 3.5 cm and height 10 cm. The hemisphere has the same radius. Find the total surface area. (Use π = 22/7)

Step 1: Total surface area = Curved surface area of cylinder + Base area of cylinder + Curved surface area of hemisphere Step 2: Curved surface area of cylinder = 2πrh = 2 × (22/7) × 3.5 × 10 Step 3: = 2 × (22/7) × 35 = 2 × 22 × 5 = 220 cm² Step 4: Base area of cylinder = πr² = (22/7) × (3.5)² Step 5: = (22/7) × 12.25 = 22 × 1.75 = 38.5 cm² Step 6: Curved surface area of hemisphere = 2πr² = 2 × (22/7) × (3.5)² Step 7: = 2 × (22/7) × 12.25 = 2 × 38.5 = 77 cm² Step 8: Total surface area = 220 + 38...

Q2: A cone is placed on top of a cylinder. The cylinder has radius 5 cm and height 8 cm. The cone has the same radius and slant height 13 cm. Find the total volume and total curved surface area.

Step 1: Volume of cylinder = πr²h = (22/7) × 5² × 8 Step 2: = (22/7) × 25 × 8 = (22/7) × 200 Step 3: = 4400/7 ≈ 628.57 cm³ Step 4: For cone, slant height l = 13 cm, radius r = 5 cm Step 5: Height of cone: h² = l² − r² = 13² − 5² = 169 − 25 = 144 Step 6: h = 12 cm Step 7: Volume of cone = (1/3)πr²h = (1/3) × (22/7) × 25 × 12 Step 8: = (1/3) × (22/7) × 300 = (22 × 100)/7 = 2200/7 ≈ 314.29 cm³ Step 9: Total volume = 4400/7 + 2200/7 = 6600/7 ≈ 942.86 cm³ Step 10: Curved surface area of cylinder = 2π...

Q3: Find the volume of a frustum of a cone if the radii of the two circular ends are 5 cm and 3 cm, and the height is 6 cm. (Use π = 22/7)

Step 1: Volume of frustum = (1/3)πh(R² + r² + Rr) Step 2: Here, R = 5 cm (larger radius), r = 3 cm (smaller radius), h = 6 cm Step 3: Volume = (1/3) × (22/7) × 6 × (5² + 3² + 5×3) Step 4: Volume = (1/3) × (22/7) × 6 × (25 + 9 + 15) Step 5: Volume = (1/3) × (22/7) × 6 × 49 Step 6: Volume = (22/7) × 2 × 49 Step 7: Volume = (22/7) × 98 Step 8: Volume = 22 × 14 = 308 cm³ Final Answer: 308 cm³

Q4: A solid is made by combining a hemisphere and a cone. The hemisphere has radius 4 cm. The cone has the same base radius and height 3 cm. Find the total surface area of the solid. (Use π = 3.14)

Step 1: Total surface area = Curved surface area of hemisphere + Curved surface area of cone Step 2: Curved surface area of hemisphere = 2πr² = 2 × 3.14 × 4² Step 3: = 2 × 3.14 × 16 = 100.48 cm² Step 4: For cone, r = 4 cm, h = 3 cm Step 5: Slant height l = √(r² + h²) = √(16 + 9) = √25 = 5 cm Step 6: Curved surface area of cone = πrl = 3.14 × 4 × 5 = 62.8 cm² Step 7: Total surface area = 100.48 + 62.8 = 163.28 cm² Final Answer: 163.28 cm²

Q5: Find the volume of a composite solid formed by joining a cylinder (radius 6 cm, height 10 cm) and a cone (same radius, height 8 cm) at the cylinder's top.

Step 1: Volume of cylinder = πr²h = (22/7) × 6² × 10 Step 2: = (22/7) × 36 × 10 = (22/7) × 360 Step 3: = 7920/7 ≈ 1131.43 cm³ Step 4: Volume of cone = (1/3)πr²h = (1/3) × (22/7) × 6² × 8 Step 5: = (1/3) × (22/7) × 36 × 8 Step 6: = (1/3) × (22/7) × 288 Step 7: = (22 × 96)/7 = 2112/7 ≈ 301.71 cm³ Step 8: Total volume = 7920/7 + 2112/7 = 10032/7 ≈ 1433.14 cm³ Final Answer: 10032/7 cm³ or approximately 1433.14 cm³

Q6: A frustum of a cone has slant height 10 cm, and the radii of the two ends are 6 cm and 4 cm. Find the curved surface area of the frustum. (Use π = 22/7)

Step 1: Curved surface area of frustum = π(R + r)l Step 2: Here, R = 6 cm (larger radius), r = 4 cm (smaller radius), l = 10 cm (slant height) Step 3: Curved surface area = (22/7) × (6 + 4) × 10 Step 4: = (22/7) × 10 × 10 Step 5: = (22/7) × 100 Step 6: = 2200/7 ≈ 314.29 cm² Final Answer: 2200/7 cm² or approximately 314.29 cm²

Showing 6 of 9 questions. Visit the full page for complete solutions.