Surface Area and Volume
Surface Area and Volume
Three-dimensional mensuration gives 2 to 3 questions and is the most formula-dense topic in the paper. The recurring difficulty is not recall but choosing between total surface area, curved surface area and volume — the question wording decides, and it is easy to misread.
The Solid Figures
| Solid | Volume | Total surface area |
|---|---|---|
| Cube | a³ | 6a² |
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πr(r + h) |
| Cone | ⅓πr²h | πr(r + l) |
| Sphere | ⅔πr³ | 4πr² |
| Hemisphere | ⅔πr³ ÷ 2 | 3πr² |
Curved surface areas differ from the totals: a cylinder's is 2πrh, a cone's is πrl and a sphere's equals its total, 4πr², since it has no flat face.
Cone: l² = r² + h² — slant height by Pythagoras
The slant height is not the height. A cone question that gives the vertical height and asks for surface area always needs this step first, and skipping it produces a clean wrong answer.
✗ Curved surface area of a cone is πrh | ✓ It is πrl, using the slant height, not the vertical height
Reading the Question Correctly
| The question mentions | You need |
|---|---|
| Painting or covering a solid | Total surface area |
| A label wrapped round a tin | Curved surface area |
| Filling with water or sand | Volume |
| An open tank or pipe | Surface area minus the missing face |
| Cost per square metre | Area |
| Cost per cubic metre or per litre | Volume |
The open-vessel case is asked often: an open cylindrical tank has surface area 2πrh + πr², with only one circular face, not two.
Remember the unit conversion as well: 1 cubic metre = 1000 litres, and 1 cubic centimetre = 1 millilitre. Questions about a tank's capacity almost always end in litres.
Scaling in Three Dimensions
- If every dimension is multiplied by k, surface area becomes k² times and volume k³ times.
- Doubling the radius of a sphere makes the surface area four times and the volume eight times.
- Two similar solids with dimensions in the ratio 2 : 3 have volumes in the ratio 8 : 27.
Melting and Recasting
A whole family of questions works on one principle: when a solid is melted and recast into another shape, the volume stays the same while the surface area changes.
A sphere of radius 3 melted into a cylinder of radius 3: volume ⅔π(27) = 36π, and π(9)h = 36π, so the height is 4 units.
The same principle covers a well being dug and its earth spread as an embankment, or a metal sheet rolled into a cylinder. Once you spot the phrase melted, recast, dug or rolled, write the volume equation first and everything else follows.
पिघलाकर ढालने पर आयतन समान रहता है, पृष्ठीय क्षेत्रफल बदल जाता है।
TRE pointer: Two errors dominate this topic: using the vertical height where the slant height is required in a cone, and counting two circular faces on an open vessel. The melting-and-recasting family is the most common word-problem shape, and it always reduces to equating two volumes. Given −1/3 negative marking, underline whether the question says curved, total or volume before you pick a formula.
60-Second Recap
- Cube 6a² and a³; cuboid 2(lb+bh+hl) and lbh.
- Cylinder: curved 2πrh, total 2πr(r+h), volume πr²h.
- Cone: curved πrl, total πr(r+l), volume one-third πr²h, with l² = r² + h².
- Sphere: surface 4πr², volume four-thirds πr³.
- Scaling by k gives k² times the surface area and k³ times the volume.
- On melting and recasting, volume is conserved; 1 cubic metre = 1000 litres.
Frequently Asked Questions
What is the difference between slant height and height in a cone?
The height is the vertical distance from the base to the apex, while the slant height runs along the surface from the base edge to the apex. They are linked by Pythagoras, with slant height squared equalling radius squared plus height squared. Curved surface area needs the slant height.
How does the surface area of an open cylindrical tank differ?
It has only one circular face, so its area is two pi r h plus pi r squared, rather than two pi r into r plus h. Questions about open tanks, pipes and buckets rely on candidates applying the closed formula out of habit.
If all dimensions of a solid are doubled, what happens to volume?
The volume becomes eight times as large, since it scales with the cube of the linear factor, while the surface area becomes four times, scaling with the square. Answering that both double is the standard error in this type of question.
How do I approach melting and recasting problems?
Write down that the volume before equals the volume after, since the material is conserved. Substitute the two volume formulas and solve for the unknown dimension. The surface area changes freely and is never conserved, which some questions test directly.