Introduction
Many mensuration problems combine solids (e.g., a cone on a cylinder) or convert material from one solid to another (melting and recasting). This pattern teaches how to add/subtract volumes and use volume equality when solids are transformed - a common and high-yield exam topic.
Mastering this helps solve practical questions like tank capacities, metal casting, hollow objects, and shaded/remaining volumes.
Pattern: Combination and Conversion of Solids
Pattern
Key concept: For combined shapes, add/subtract volumes (or areas). For conversion, equate total volume before = total volume after.
Useful relations:
• Volume add/subtract: Volume(composite) = Sum(volumes of parts) - Volume(removed parts).
• Conversion: Volume(before) = Volume(after) (mass and density constant).
• Common formulas used inside problems: cylinder πr²h, cone (1/3)πr²h, sphere (4/3)πr³, cuboid lbh.
Step-by-Step Example
Question
A solid metal sphere of radius 9 cm is melted and recast into identical cones each with base radius 3 cm and height 8 cm. How many full cones can be made?
Solution
-
Step 1: Compute volume of the sphere.
Sphere volume = (4/3)πR³. With R = 9 cm: V_sphere = (4/3)π × 9³ = (4/3)π × 729 = 972π. -
Step 2: Compute volume of one cone.
Cone volume = (1/3)πr²h. With r = 3 cm, h = 8 cm: V_cone = (1/3)π × 3² × 8 = (1/3)π × 9 × 8 = 24π. -
Step 3: Use conversion equality.
Number of cones = V_sphere ÷ V_cone = (972π) ÷ (24π) = 972 ÷ 24 = 40.5. -
Step 4: Final Answer.
Only full cones count → 40 full cones can be made (half-cone leftover). -
Quick Check:
40 × 24π = 960π which is less than 972π; one more cone would require 24π more which exceeds sphere volume ✅
Quick Variations
1. A cylinder with a cylindrical cavity (hollow pipe): subtract inner cylinder volume from outer one.
2. A solid made of a cone on top of a cylinder: add cone and cylinder volumes.
3. Metal from multiple small solids combined to form a larger solid: sum small volumes then equate to larger shape.
4. Shaded region problems: subtract inner solid/area from outer to get remaining material.
5. Conversion with loss/gain: if given percentage loss, multiply before equating (e.g., usable volume = (1 - loss%) × initial volume).
Trick to Always Use
- Step 1 → Write exact formulas for each part (no approximations early) and keep π symbolic until cancellation is possible.
- Step 2 → When converting, cancel common factors (π, 1/3, etc.) to simplify before numeric computation.
- Step 3 → For "how many" questions, divide total available volume by single-item volume and take the integer part (floor) for full items.
- Step 4 → Check feasibility (e.g., ensure source volume ≥ required volume for at least one item) and consider material loss if specified.
Summary
Summary
Combination and conversion problems reduce to careful volume bookkeeping:
- Add volumes for joined parts; subtract for cavities or removed parts.
- For recasting, equate total initial volume to total final volume (apply loss factor if given).
- Keep units consistent and delay using π's numeric value until it cancels or final numeric answer is needed.
- For integer counts, always take the floor of the division result for full items; report leftovers if asked.
Hint: For identical shapes: number = (linear scale)³ = (R/r)³.
Common Mistakes: Using diameters instead of radii when cubing.