Topic · Mensuration (3-D)

Volume and surface area: picking the right solid, and the right height

The one-third that separates a cone from a cylinder, why slant height and vertical height are not interchangeable, and what happens to volume when every length is doubled.

6 min readUpdated 5 September 2026

Three-dimensional mensuration questions look like formula recall and are not. The formula sheet is on the desk; what the question tests is whether you can name the solid you are looking at, identify which of its lengths the formula actually wants, and — when two solids are related — carry a scale factor through the right power.

You will usually meet one of three shapes of question: compute a volume or surface area from given dimensions; compare two related solids; or work backwards from a volume to a missing length.

Misconception 1: the formula of a lookalike

A cone and a cylinder on the same base with the same height look similar enough in a diagram that the wrong formula does not feel wrong. So do a pyramid and a prism. The difference is a factor of $\tfrac{1}{3}$ , and it is not decoration — a solid that tapers to a point holds a third of the one that does not.

V_{\text{cylinder}} = \pi r^2 h \qquad V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h

The habit that fixes this is to say out loud what the solid does at the top. If it comes to a point, there is a one-third. If it does not, there is not.

Misconception 2: feeding the wrong length into a surface area

The curved surface area of a cone uses the slant height, the distance from the rim to the apex along the surface. The volume uses the vertical height, the perpendicular distance from the base to the apex. They are two different segments of the same right-angled triangle, and a question that gives you one usually expects you to find the other.

The second half of this trap is the base. "Curved surface area" excludes the circular base; "total surface area" includes it. For a cylinder, "total" usually means two circles, not one. Read which word the question used before you add anything.

Misconception 3: one scale factor for everything

If two solids are similar with lengths in the ratio $k$ , their surface areas are in the ratio $k^2$ and their volumes in the ratio $k^3$ . Doubling every length multiplies the volume by $8$ , not by $2$ . And the rule only applies when the solids are genuinely similar — a cylinder whose radius is doubled but whose height is unchanged is not similar to the original, so no single $k$ exists and the whole shortcut is unavailable.

A worked example

A solid cone has base radius $3$ cm and vertical height $4$ cm. Find its volume and its total surface area.

The slant height comes first, from the right-angled triangle formed by the radius, the vertical height and the slant:

\ell = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \text{ cm}.

Then

V = \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3}\pi (3)^2 (4) = 12\pi \text{ cm}^3,

\text{total surface area} = \pi r \ell + \pi r^2 = \pi(3)(5) + \pi(3)^2 = 15\pi + 9\pi = 24\pi \text{ cm}^2.

Two wrong turns are worth naming. Dropping the one-third gives $36\pi$ — a clean-looking answer that is exactly three times too big. Putting the vertical height into the curved-surface formula gives $\pi(3)(4) = 12\pi$ instead of $15\pi$ , which is wrong by a factor small enough that nothing about the number looks suspicious.

Now suppose a second, similar cone has base radius $6$ cm. Its length scale factor is $k = 2$ , so its volume is $2^3 \times 12\pi = 96\pi$ cm³ and its total surface area is $2^2 \times 24\pi = 96\pi$ cm². The two numbers coinciding here is a coincidence of these particular dimensions — a useful reminder that a matching number is not a check.

Check yourself

Two questions before you write the formula down: "does it come to a point?" and "which height does this formula want?" Then, if a second solid is involved, "are these actually similar, and is this a length, an area or a volume?"

The drill below rebuilds the solid from new dimensions every time, so the formula choice is a fresh decision each question.

Now do some

A drill session on Mensuration (3-D) — fresh questions each time, generated from the same syllabus objectives this guide explains, and machine-checked twice before you see them.

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Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.