Mathematics · Class 10
Surface Areas and Volumes
Frustum, combination solids, and conversion between 2D and 3D measures.
3 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Class 10 Surface Areas and Volumes extends Class 9 solids to combinations of shapes and the frustum of a cone. You calculate TSA, CSA, and volume when solids are joined or hollowed.
Combination of Solids
- Add volumes when solids are joined (no overlap)
- For surface area, subtract hidden faces where solids meet
- Common combos: cylinder + hemisphere, cone + cylinder
- Hollow solids: subtract inner volume from outer
Frustum of Cone
Volume
V = (πh/3)(R² + Rr + r²)
CSA
πl(R + r), where l = slant height
Slant height
l = √[h² + (R − r)²]
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Tent (cone + cylinder)
Tent: cylindrical base r = 7 m, h = 2 m; conical top same r, height 3 m. Find total volume.
- 1V_cyl = π × 49 × 2 = 98π
- 2V_cone = ⅓π × 49 × 3 = 49π
- 3Total = 147π m³
Answer
147π m³
Example 2: Frustum volume
Frustum: R = 6 cm, r = 3 cm, h = 4 cm. Find volume.
- 1V = (π×4/3)(36 + 18 + 9)
- 2= (4π/3)(63) = 84π cm³
Answer
84π cm³
Key Points to Remember
- ✓ Hidden surfaces are not counted in TSA of combined solid
- ✓ Frustum is cone minus small cone
- ✓ Use consistent units throughout
Exam Tips
- • Sketch combined solid and mark shared faces
- • Write individual volumes before adding
- • Frustum problems: find l before CSA
Practice MCQs — Surface Areas and Volumes
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
Volume of combined solids is: