Similar figures questions look deceptively simple. You are given two cones, two triangles or two water tanks, one length, one area, and asked for the missing value. Yet every year students lose easy marks here because they multiply when they should square, or square when they should cube.
The good news: this topic in the Singapore/Cambridge O-Level E Math syllabus runs on one single rule. Once you drill it, these are among the fastest marks on Paper 1 and Paper 2.
The one rule you actually need
If two figures are similar with length scale factor k, then:
- Length ratio = k
- Area ratio = k²
- Volume ratio = k³
Write it as a ladder in your notes:
| Quantity | Ratio |
|---|---|
| Any length (side, radius, height, slant height, perimeter, circumference) | k |
| Any area (surface area, cross-section, base area) | k² |
| Any volume (or mass, if the material is the same) | k³ |
Two things students forget:
- Perimeter behaves like a length, not an area. Perimeter ratio is k, not k².
- Mass and weight behave like volume when the objects are made of the same material with the same density. Many O-Level questions about solid metal shapes or model statues rely on this.
Finding k correctly
Always set up k as new over old, or consistently big over small. Mixing directions mid-question is the single most common error.
If you are given areas and need lengths, take the square root. If given volumes, take the cube root.
- Area ratio 9 : 25 → k = 3 : 5
- Volume ratio 8 : 27 → k = 2 : 3
- Volume ratio 1 : 2 → k = 1 : ∛2 ≈ 1 : 1.26
That last one is worth noting. Ratios do not have to be nice. If a question says one tank holds twice the volume of another, the lengths are in ratio 1 : 1.26, not 1 : 2.
Worked example 1: Two similar cones
Two similar cones have heights 6 cm and 9 cm. The smaller cone has curved surface area 45 cm² and volume 84 cm³. Find the curved surface area and volume of the larger cone.
Step 1. k = 9/6 = 3/2 (large over small).
Step 2. Area: multiply by k².
Curved surface area = 45 × (3/2)² = 45 × 9/4 = 101.25 cm²
Step 3. Volume: multiply by k³.
Volume = 84 × (3/2)³ = 84 × 27/8 = 283.5 cm³
Notice you never needed the radius or slant height. That is the whole point of the ratio method, and examiners reward it. If you try to reverse engineer the radius using πr²h/3, you will burn three minutes and risk rounding errors.
Worked example 2: Working backwards
Two similar solid statues are made of the same bronze. The smaller has mass 4 kg, the larger 13.5 kg. If the smaller statue is 20 cm tall, find the height of the larger.
Mass behaves like volume, so:
k³ = 13.5 / 4 = 3.375
k = ∛3.375 = 1.5
Height = 20 × 1.5 = 30 cm
Write the line k³ = 13.5/4 explicitly. Method marks are given for showing that you have used a cube ratio, even if the arithmetic slips.
Worked example 3: The frustum trap
A cone of height 12 cm is cut parallel to its base, 4 cm from the apex, forming a small cone and a frustum. Given that the whole cone has volume 216 cm³, find the volume of the frustum.
The frustum is not similar to the cone. Only the small cone and the whole cone are similar.
k = 4/12 = 1/3
Small cone volume = 216 × (1/3)³ = 216 × 1/27 = 8 cm³
Frustum volume = 216 − 8 = 208 cm³
Always go via the two similar cones, then subtract. This is a standard Paper 2 structure and appears with surface areas too.
Mark scheme phrasing that earns marks
When a question asks you to explain or prove similarity first, use the accepted wording:
- "Angle ABC = angle ADE (corresponding angles, BC parallel to DE)"
- "Angle A is common"
- "Therefore triangle ABC is similar to triangle ADE (AA)"
Order your letters so that corresponding vertices match. Writing "triangle ABC similar to triangle AED" when the correspondence is ADE can cost you the mark even if your working is right.
Common mistakes to eliminate this week
- Using k instead of k² for surface area of a 3D solid.
- Using k² for mass or capacity instead of k³.
- Inverting k halfway through a multi-part question.
- Rounding k too early. Keep the exact fraction or surd until the final answer, then round to 3 significant figures.
- Assuming a frustum is similar to the original cone.
A five-minute drill
Take any past paper question with two similar solids. Cover the numbers and ask yourself only one question: is this a length, an area or a volume? Then write k, k² or k³ before touching your calculator. Do ten of these and the topic is done.
If ratios, mensuration or any other E Math topic still feels shaky, our tutors have sat these exact papers recently and know where the marks hide. Start a free trial class with us on WhatsApp and bring your toughest question along.