Enlarge a photo 2× in each direction — how much more ink does it need?
Similarity is scaling that preserves shape — lengths change by the ratio, areas by its square.
Experiment
Hands-on experiment
🔮 Predict first — enlarge a photo 2× in each direction. How much more ink?
🖼 Change the zoom
Count the cells yourself — what happens to the width vs the cell count (area)?
width ×1
cells ×1
2×2 = 4
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Drawing maps, building models, printing photos — humanity kept hitting one question: when size changes, what is preserved and what transforms?
The answer splits in two: angles and ratios are preserved (so shape survives), while lengths, areas, and volumes scale by k, k², k³ — one similarity ratio governs everything.
The rule became a surveying tool. A stick's shadow and a tree's shadow make similar triangles — Thales measured the pyramids this way, and the same principle grows into trigonometry.
Insight
Insights from the video
“Similarity is the eye that ignores size.”
Maps, blueprints, miniatures, zoomed screens — all lean on 'different size, same information'. The ratio lab's 'relationship with size erased' arriving in the world of shapes.
“Why elephants can't jump like ants — also similarity.”
Scale a body by k: muscle strength grows with cross-section (k²), weight with volume (k³) — the bigger you are, the heavier per strength. The square-cube law shapes biology.
Misconception
Common misconceptions
Scale by 2 and everything doubles.
Only lengths double. Areas quadruple (2²), volumes octuple (2³) — each dimension has its own factor. That's why photo prints get pricier faster than their size.
Different sizes make different shapes.
To the similarity eye they're the same shape — all angles and all side ratios agree. That maps equal real land and models equal real things rests on similarity.
Formula
Writing it as math
What the photo enlargement confirmed, in mathematical language.
Defining similarity
Corresponding angles equal; corresponding sides in one constant ratio — same shape, different size.
Factor per dimension
One ratio k governs every dimension. A 2× photo needs 4× the ink (area).
Shadow surveying
Same moment, same sun angle → similar triangles. Equal side ratios measure heights you cannot climb.
In Real Life
Where you meet it in real life
Maps and scales
A 1:25,000 map is similar to the land. 4cm on paper = 1km on the ground — one ratio translates every distance.
Models and design
1/100 architecture models, 1/24 car kits — models carry the real thing's information because of similarity. The trap: weight scales by (1/100)³.
Prints and screens
Enlarging a print raises ink and price with area (k²). A 4×6 and an 8×12 differ by ratio 2 — and cost logic of 4.
The size law of life
The square-cube law: volume (weight) outgrows cross-section (strength) as bodies scale — why ants lift 50× themselves and elephants never jump.
Try Yourself
Test yourself
Q1A 1m stick casts a 1.5m shadow. How tall is a tree with a 12m shadow?Show answer ▾
Equal height/shadow ratios: height = 12 × (1/1.5) = 8m — precisely how Thales measured the pyramid.
Q2Two triangles have similarity ratio 3. Their area ratio?Show answer ▾
3² = 9. Lengths scale by the ratio, areas by its square — both directions stretch 3×, giving 9.
Q3A 1/24-scale model car has what fraction of the real volume?Show answer ▾
(1/24)³ = 1/13,824 — about one fourteen-thousandth. Length shrinks 24×, but volume (material) over ten-thousand-fold — the power of the cube.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
Related video
Connection
Concepts connect
Previous concept
Leads to next
Pythagorean Theorem
Armed with the eye that compares shapes, meet the astonishing relation hiding in right triangles.
Go to the Pythagorean Theorem lab →Related
Labs worth exploring together
Related lab
Ratios
Similarity is 'the relationship with size erased', applied to shapes — ratios are the root.
Go to the Ratios lab →Related lab
Trigonometric Ratios
Equal angles give equal ratios (similarity) → per-angle ratio tables (trig).
Go to the Trigonometric Ratios lab →