Melting and recasting problems have one golden rule: the VOLUME of metal never changes shape-to-shape, even though the surface area always does.
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Volume measures how much a solid can HOLD; surface area measures how much material would COVER its outside.
📊 The core formulas
| Solid | Volume | Total surface area |
|---|---|---|
| Cube (side s) | s³ | 6s² |
| Cuboid (l, w, h) | l × w × h | 2(lw + wh + hl) |
| Cylinder (radius r, height h) | πr²h | 2πr(r + h) |
Worked example: a cylinder with r=7, h=10 (π=22/7): volume = (22/7)×49×10 = 1540; total surface area = 2×(22/7)×7×(7+10) = 748.
A cone's volume is exactly ONE THIRD of a cylinder with the same base and height — this connection is worth remembering.
📌 Cone and sphere formulas
Cone volume = ⅓ × πr² × h. For r=7, h=6: volume = ⅓×(22/7)×49×6 = 308.
Cone slant height l = √(r² + h²), used for curved surface area = πrl.
Sphere volume = 4/3 × πr³. Sphere surface area = 4πr².
Worked example — slant height: a cone with r=7 and h=24 has slant height l=√(7²+24²)=√625=25. Its curved surface area = πrl = (22/7)×7×25 = 550.
When a solid is melted and recast into a new shape, only the VOLUME stays exactly the same — set the two volume formulas equal to each other.
Worked example: a metal sphere of radius 6 cm is melted and recast into a cylinder of radius 3 cm. Sphere volume = (4/3)π×216 = 288π. Setting this equal to the cylinder's volume πr²h = 9πh gives h = 288÷9 = 32 cm.
12 questions on volume, surface area, and melting & recasting.
