Volume fills, surface area covers
The two quantities answer different questions and are measured in different units. Volume is how much fits inside, in cubic units. Surface area is how much material would wrap around the outside, in square units.
Deciding which a question wants is the first step, and the wording gives it away. "How much water will it hold" is volume; "how much paint to cover it" is surface area. So is "how much card is needed to make it" — a net question in disguise.
Compare the cylinder and the cone at the same size. The cone holds exactly one third of the cylinder that would contain it — which is worth remembering as a check rather than a separate formula.
Prisms: one rule covers all of them
A prism is a solid with the same cross-section all the way along its length. A cuboid is a prism with a rectangular cross-section; a cylinder is a prism with a circular one; a triangular prism has a triangular one.
That gives a single formula for all of them: volume = area of cross-section × length. Rather than memorising a separate formula for each solid, find the area of the face that repeats and multiply by how far it extends.
Why the curved surface of a cylinder is 2πrh
Cut the label off a tin and unroll it. It is a rectangle: its height is the height of the tin, and its width is the distance all the way round — the circumference, 2πr. So the curved area is 2πrh, and adding the two circular ends of πr² each gives the total. Deriving it this way makes the formula impossible to misremember.
Cones, pyramids and spheres
These are not prisms — the cross-section shrinks as you move up — so the prism rule does not apply. Their formulas are given on the exam paper, but knowing what each symbol means is not.
The point that costs marks is the difference between the vertical height h and the slant height l of a cone. The volume uses h, the perpendicular distance from base to apex. The curved surface area uses l, measured along the sloping side. They are connected by Pythagoras, and a question giving one while requiring the other is entirely standard.
- r
- radius of the base
- h
- perpendicular height from base to apex
- l
- slant height along the sloping surfacealways the longest of the three
A cone has base radius 5 cm and vertical height 12 cm. Find its volume and its total surface area. Take π = 3.142.
- Volume
= ⅓πr²h = ⅓ × 3.142 × 25 × 12 = 314.2cm³.The vertical height goes into the volume. - For the curved surface, first find the slant height:
l² = 5² + 12² = 169, sol = 13.A 5-12-13 triangle — examiners choose these numbers so the root is exact. - Curved surface
= πrl = 3.142 × 5 × 13 = 204.2cm².The slant height, not the vertical one. - Base
= πr² = 3.142 × 25 = 78.6cm²."Total" surface area includes the base; "curved" does not. - Total
= 204.2 + 78.6 = 282.8cm².
Volume 314.2 cm³; total surface area 282.8 cm²
Read whether the base is included
A cone-shaped hat has no base, so only the curved surface is wanted. A solid cone standing on a table has one. Questions say "curved surface area" or "total surface area", and the two differ by πr² — enough to lose the final mark. The same applies to an open cylinder such as a pipe or a tin without a lid.
Composite solids and the effect of scaling
A shape made of two solids joined together is handled by adding volumes, exactly as with areas. Surface area needs more care: where the two meet, the joining faces are inside the solid and are not part of the surface.
And the scaling rule from similar figures applies to solids too. Multiply every length by k and the surface area multiplies by k² while the volume multiplies by k³. That is why a small model needs far less material than its scale suggests, and why large animals have proportionally thicker legs than small ones.
Before you leave this chapter
- Volume fills (cubic units); surface area covers (square units). Read which is wanted.
- Every prism: volume = cross-section area × length.
- A cylinder's curved surface unrolls into a rectangle 2πr by h.
- Cone volume uses the vertical height; curved surface uses the slant height, with l² = r² + h².
- Lengths × k gives areas × k² and volumes × k³.
Nets, and why they make surface area easy
A net is the flat shape that folds up into a solid. Drawing one turns a surface-area question into an ordinary area question, because every face becomes a flat shape whose area you already know how to find.
A cuboid unfolds into six rectangles, in three matching pairs. A cylinder unfolds into two circles and one rectangle whose width is the circumference. A cone unfolds into a circle and a sector. Sketching the net first is the surest way to avoid counting a face twice or missing one entirely.
| Solid | Net consists of | Total surface area |
|---|---|---|
| Cuboid | 6 rectangles in 3 pairs | 2(lw + lh + wh) |
| Cylinder | 2 circles + 1 rectangle | 2πr² + 2πrh |
| Cone | 1 circle + 1 sector | πr² + πrl |
| Triangular prism | 2 triangles + 3 rectangles | 2(½bh) + the three faces |
| Square pyramid | 1 square + 4 triangles | b² + 4(½ × b × slant) |
Counting the faces is the whole method
Most lost marks in surface area come from a missing or duplicated face, not from arithmetic. Sketch the net, label each piece with its dimensions, find each area, and add. It takes a minute longer than trying to hold the solid in your head and it is very much more reliable — particularly for an open container, where one face is deliberately absent.