Surface Areas and Volumes

Why This Chapter Matters

Mensuration is where geometry meets physical measurement. A container can be painted, filled, wrapped, covered or melted, and each question asks for a different quantity.

The biggest skill is not memorising formulas; it is deciding what surface or space the problem is actually asking about. Painting a closed cylinder requires total surface area. Labelling only the curved side requires curved surface area. Filling a tank requires volume.

Draw the object, mark dimensions and identify whether the question is about covering, exposed surface or capacity before choosing a formula.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Area of rectangles and circles
• Volume concept
• Pythagoras theorem
• Unit conversion

These are the tools this chapter reuses and deepens.

1. Surface Area vs Volume

Surface area measures the amount of material needed to cover surfaces. It answers questions about paint, sheet metal, wrapping and labels.

Volume measures three-dimensional space occupied. Capacity is closely related to volume and is often expressed in litres or millilitres.

Never attach cubic units to surface area or square units to volume.

2. Cuboid and Cube

A cuboid has dimensions l, b and h. Its total surface area comes from three pairs of equal rectangles.

Lateral surface area excludes top and bottom. A cube is the special case l=b=h=a, so the same ideas simplify to square-face formulas.

3. Cylinder

A cylinder has two circular bases and one curved surface.

If the curved surface is cut vertically and opened, it becomes a rectangle whose length is the circumference 2πr and whose width is h. This explains why curved surface area is 2πrh rather than leaving it as a formula to memorise.

4. Cone

A cone has radius r, vertical height h and slant height l.

The slant height forms a right triangle with r and h, so l²=r²+h². Curved surface area uses l because the surface slopes, while volume uses perpendicular height h.

Confusing l and h is one of the most frequent mensuration errors.

5. Sphere and Hemisphere

A sphere has no flat face. Its whole surface area is 4πr² and its volume is 4/3 πr³.

A hemisphere is half a sphere in volume, but its total surface area is not simply half the sphere's surface area because the hemisphere also has a circular base. Therefore CSA=2πr² but TSA=3πr².

6. Unit Conversion

Convert all dimensions before substitution. Do not mix metres and centimetres inside one formula.

Useful capacity links are 1 cm³ = 1 mL and 1000 cm³ = 1 L. Remember that area conversions square the linear factor while volume conversions cube it.

7. Practical Interpretation

Words such as open, closed, hollow, filled, painted outside or melted and recast determine which quantity is needed.

In recasting questions, volume is conserved if no material is lost. Surface area generally changes because the shape changes.

Visualise the Core Idea

Choosing the correct dimension and exposed surface is the main skill in mensuration.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary