Similar Shapes and Scale Factors

La méthode, étape par étape

1. 1

   Confirm the shapes are similar

   equal angles and a constant ratio between corresponding sides

   Similarity requires both conditions, though for triangles equal angles is enough to guarantee proportional sides. Checking which sides correspond matters — matching the wrong pair gives a scale factor that is wrong from the start.

2. 2

   Find the length scale factor

   corresponding sides 6 cm and 9 cm → k = 9 ÷ 6 = 1.5

   Divide a length on the second shape by the corresponding length on the first. Getting the direction right matters: going the other way gives 0.667, which is correct for the reverse comparison and wrong for this one.

3. 3

   Scale areas by k²

   k = 1.5 → area scale factor 1.5² = 2.25

   Area is two lengths multiplied, so both are scaled and the factor is squared. A shape with sides 1.5 times longer has 2.25 times the area, not 1.5 times — this is the topic's defining insight.

4. 4

   Scale volumes by k³

   k = 1.5 → volume scale factor 1.5³ = 3.375

   Volume is three lengths multiplied, so the factor is cubed. Doubling every length of a solid multiplies its volume by eight, which is genuinely surprising and is examined precisely because it is.

5. 5

   Work backwards by rooting

   areas 20 and 45 cm² → k² = 2.25 → k = 1.5

   Given an area ratio, square root it to recover the length scale factor; given a volume ratio, take the cube root. Using the area ratio directly as a length factor is the most common error in reverse questions.

Why area squares and volume cubes

A rectangle 2 by 3 has area 6. Double every length and it becomes 4 by 6, area 24 — four times as much, not twice. Both dimensions grew, so the growth applied twice. For a solid, three dimensions grow and the factor applies three times.

Students find this genuinely counter-intuitive and it is worth doing the arithmetic in front of them rather than asserting the rule. Once someone has computed 4 × 6 = 24 themselves, the k² rule stops being something to remember.

• Similar → equal angles, proportional sides
• Length scale factor k
• Area scale factor k²
• Volume scale factor k³
• Reverse: square root an area ratio, cube root a volume ratio

Similar and congruent

Congruent shapes are identical in size and shape. Similar shapes have the same shape but may differ in size — so every congruent pair is also similar, with a scale factor of 1, but the reverse does not hold.

The everyday meaning of 'similar' is looser than the mathematical one and causes confusion. In mathematics it is a precise condition, not a description of shapes that look alike.

Where this rule bites in the real world

Doubling the linear size of a container multiplies its capacity by eight, which is why a pizza twice the diameter is four times the food and priced accordingly. It is also why models and full-size objects cannot share the same proportions of strength and weight.

These examples matter for exams as well as interest. Questions about paint coverage, tin capacity and model scaling are standard, and they are all this one rule applied to a context.

How we teach similar shapes

We start with the arithmetic demonstration — compute the area of a rectangle, double the sides, compute again — before stating any rule. The k² result is memorable when a student has produced it and forgettable when they have been told it.

We also drill the reverse direction separately. Recovering a length factor from an area ratio requires a square root, and students who have only practised the forward direction routinely skip it.