∽ · Similarity
Lengths scale by the ratio; areas by its square.
Enlarge a photo 2× in each direction — how much more ink does it need?
Distinguish length ratios from area ratios.
Thales, who measured the Great Pyramid's height with nothing but a stick's shadow.
Open the immersive version →A reading path containing this lesson: Measure without reaching · 1/3
Why does this question matter?
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.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
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 the idea from another angle
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.
Now read the definitions with their conditions
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.
Make one more distinction
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.
Where else can this idea help?
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.
Explain it in your own words
Try another question
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.
What question comes next?
A reading path containing this lesson: Measure without reaching · 1/3
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 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 →