Similar Triangles Lab

Set the three sides, then drag the scale factor. The angles hold still, the sides follow k, and the area follows k × k.

Worked example — Similar Triangles Lab

Sides	3, 4, 5 → 6, 8, 10
Perimeter	12 → 24 (×2)
Area	6 → 24 (×4)
Angles	36.87°, 53.13°, 90.00° — unchanged
Copies of the small triangle that fit	4

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Twice as tall is four times the paint 🖖

This is the most expensive arithmetic mistake most people make, and the arithmetic is not the part that goes wrong. Doubling a shape does not give you two of it. It gives you four. A pizza of twice the diameter is four pizzas. A photograph at twice the width needs four times the pixels. A model at 1:2 needs a quarter of the material, not half. Set k to a whole number and the tool stops asserting it: the big triangle is drawn cut into the copies it actually contains, and you can count them.

Why the count is exactly k² 🖖

Take the scaled triangle and mark every side into k equal parts, then join the marks with lines parallel to the three sides. The result tiles perfectly, and the tiles are all congruent to the original — k² of them, k of which sit along the bottom row. Nothing about that argument mentions what shape the triangle was, which is why the thin sliver preset gives 16 just as the 3-4-5 does. Euclid states the general result as Elements VI.19: “similar triangles are to one another in the duplicate ratio of the corresponding sides” — and “duplicate ratio” is his phrase for the square of the ratio.

This is the measurement Thales is supposed to have made 🖖

A stick and its shadow make one triangle; a tree and its shadow make another. The sun is far enough away that its rays arrive parallel, so the two triangles have equal angles and therefore proportional sides — and a height nobody can reach becomes three lengths anybody can pace out. That is the whole method behind measuring a pyramid by its shadow, and it is the same square you fill in on the cross-multiplication grid: 3 is to 4 as the tree is to 20. Similar triangles are what makes that grid legitimate rather than a trick.

Problem solved in full

How many copies of the original 3-4-5 triangle fit inside the enlargement 5 steps
Double every side of a 3-4-5 triangle and the area does not double. Work out what each quantity does, and how many copies of the original fit inside the enlargement.
1. Scaling all three sides by the same factor keeps the triangle similar, and 6-8-10 still satisfies Pythagoras — as it must, since the relation is homogeneous in the sides.
  
  $(3, 4, 5) \times 2 (6, 8, 10), 6^{2} + 8^{2} = 100 = 1 0^{2}$
2. The perimeter is a sum of lengths, so it scales by the same factor. Twelve becomes twenty-four.
  
  $P : 12 \to 24 = k P, A : 6 \to 24 = k^{2} A$
3. The area is a product of two lengths, so it scales by the factor squared. Six becomes twenty-four — the same number as the perimeter here, which is a coincidence of these particular sides and a good reason not to reason from one example.
4. The angles do not move at all. They depend only on ratios of sides, and scaling leaves every ratio alone — which is what similar means.
  
  $tan^{- 1} \frac{3}{4} = 36.8 7^{\circ}, tan^{- 1} \frac{4}{3} = 53.1 3^{\circ}, 9 0^{\circ} — unchanged$
5. So four copies of the original fit inside the enlargement, and you can see it: the doubled triangle cuts into four triangles congruent to the original.
  
  $\frac{A ^{'}}{A} = k^{2} = 4 ⟹ 4 copies tile the enlargement$
Answer

The tool prints sides 6, 8, 10, perimeter 12 → 24, area 6 → 24, unchanged angles, and 4 copies fitting. The rule is one line: length scales as k, area as k², volume as k³, and angle as k⁰ — an exponent equal to the dimension of the thing being measured. That is why doubling a recipe's tin gives four times the base area and the cake fails, and why an animal twice as tall puts eight times the weight on four times the bone cross-section, so the stress in its legs doubles and it cannot simply be a larger copy of itself. Set the factor to 3 and the area goes up ninefold.

$P \propto k, A \propto k^{2}, V \propto k^{3}, angles \propto k^{0}$

Learning path

The triangle rules, and where they come from

Leads to Pythagoras the scale factor — that enlarging a shape multiplies every length by k and moves no angle at all.

References (3)

Equal angles give proportional sides: Euclid, Elements, Book VI, Proposition 4 — "In equiangular triangles the sides about the equal angles are proportional." David E. Joyce's edition, Clark University.
The area result, and the phrase "duplicate ratio": Euclid, Elements, Book VI, Proposition 19 — "Similar triangles are to one another in the duplicate ratio of the corresponding sides." David E. Joyce's edition, Clark University.
Thales and the shadow measurement: J. J. O'Connor and E. F. Robertson, "Thales of Miletus", MacTutor History of Mathematics Archive, University of St Andrews — which sets out the ancient reports of the pyramid measurement and their disagreements.

By the 42Math team · Updated 2026-08-11

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

Example problems

Double it - Sides 3, 4, 5 become 6, 8, 10. The perimeter doubles, 12 to 24 — but the area goes from 6 to 24, four times over, and the big triangle is drawn holding exactly 4 copies of the small one.
Stick and tree - A stick 3 m tall with a 4 m shadow, and a tree whose shadow is 20 m. Same sun, same triangle, so the tree is 15 m. Its area, though, is 25 times the stick triangle, not 5.
Equilateral - All three angles stay at 60° whatever k does. Sides 6 become 9 and the perimeter 18 becomes 27, while the area is multiplied by 2.25.
Shrink by half - Shrinking is the same rule running backwards. Halving every side halves the perimeter, 48 to 24, and quarters the area, 96 to 24.
A thin sliver - Shape means angles, not looks. This sliver scales exactly like a comfortable triangle: ×4 on the sides, ×16 on the area, and 16 copies fit inside.