Similarity
“Similarity is 'same shape, different size' — that size difference is the scale factor.”
The formula
a : a' = b : b' = c : c' = kHow to read it: the ratios of corresponding sides are all equal — that constant ratio k is the scale factor
- a, b, c / a', b', c'
- — the corresponding sides of the two figures
- k
- — the ratio of corresponding side lengths (the scale factor)
The hook
Even at different sizes, 'the same shape' is something math captures precisely with a single ratio.
In plain words
Two figures being similar means all corresponding angles are equal and all corresponding sides are the same multiple (k times) of each other.
The intuition
Think of zooming a photo in or out. Every part grows or shrinks by the same factor, so the shape stays put. That 'one steady factor' is the scale factor k. The angles hold fixed no matter the zoom.
How it's built
It splits into two axes — ① all corresponding angles equal (shape preserved), ② all corresponding side ratios equal to k (only size changes). And crucially, if length scales by k, area scales by k².
Example
Two triangles with scale factor 2: if a side of the small one is 3, the matching side of the large one is 6. The area grows not by 2 but by 2² = 4.
Common mistake
A scale factor of 2 does NOT mean the area doubles. Length scales by k, area by k², volume by k³. People often miss this 'power of k' difference.
Where it's used
Map scales, scale models versus the real object, measuring a tree's height from its shadow, zooming a screen's resolution — anywhere size changes but shape is preserved.
Where it came from
Thales is said to have measured the height of a pyramid using the ratio of shadows — the oldest recorded use of similarity.
Prerequisites
Quick check
For two figures with scale factor 3, what is the ratio of their areas?
- 3
- 6
- 9✓
- 27
Practice
Two triangles have scale factor 2 : 3. If a side of the small triangle is 6, what is the corresponding side of the large one?
Answer: 9
- Corresponding sides keep the constant ratio 2 : 3
- 6 : x = 2 : 3 → 2x = 18 → x = 9
Takeaway: Corresponding sides always share the same ratio — one proportion solves it.
For two figures with scale factor 1 : 4, what is the ratio of their areas?
Answer: 2
- The area ratio is the square of the scale factor
- 1² : 4² = 1 : 16
Takeaway: Area is the scale factor squared — 4 times the length is 16 times the area.
For two cubes with scale factor 1 : 2, what is the ratio of their volumes? Write on your notepad why.
Answer: undefined
- Volume is the cube of the scale factor
- 1³ : 2³ = 1 : 8
- Because length (1D) scales by k, area (2D) by k², volume (3D) by k³
Takeaway: Each step up in dimension raises the scale factor's exponent by one.