How to Find the Scale Factor Between Two Similar Triangles

Scale factors run drawings, models and enlargements. A site plan at 1:500, a shop drawing blown up for a detail, a truss profile stretched to a wider span: each is a similar figure, the same shape at a different size, and the scale factor is the single number relating them. For triangles the factor is found by one division, provided you divide corresponding sides, and it is verified by two more.

Corresponding sides, and how to spot them

Similar triangles have equal angles in matching order, and the sides that face equal angles correspond. In practice the sorting trick is faster than chasing angles: list each triangle from shortest side to longest, and match by rank. Shortest with shortest, middle with middle, longest with longest. For 6, 8, 10 against 9, 12, 15, that pairs 6 with 9, 8 with 12, and 10 with 15.

Then divide each pair. If all three quotients agree, the triangles are similar and the shared quotient is the scale factor. If they disagree, the triangles are not similar and no single factor exists, which is a finding, not a failure. A supposed enlargement whose ratios read 1.5, 1.5 and 1.42 has been distorted, and the mismatched pair tells you where.

Three ratios, one factor

The diagram draws the 6, 8, 10 right triangle beside its enlargement and lists all three divisions. The bottom card carries the fact people forget: the areas are 24 and 54, and 54/24 = 2.25, which is 1.5 squared, not 1.5.

A worked check you can repeat on paper

A drawing shows a gusset plate as a triangle with sides 60, 80 and 100 mm. The fabricated plate must have its longest side at 150 mm. The factor is 150/100 = 1.5, so the other sides cut at 60 × 1.5 = 90 mm and 80 × 1.5 = 120 mm. Confirm the ratios: 90/60 = 1.5 and 120/80 = 1.5. Consistent, so the enlarged plate is the same shape, not a distortion.

Now the material takeoff, where the square law earns its keep. The drawing plate has area ½ × 60 × 80 = 2400 mm², since 60, 80, 100 is right angled. The fabricated plate is not 1.5 times that but 1.5² = 2.25 times: 5400 mm². Check directly: ½ × 90 × 120 = 5400. Plate mass, paint coverage and blank size all follow the 2.25, not the 1.5.

One step further for completeness: volumes and masses of fully scaled solids follow k³. A 1.5 scale casting of the same shape weighs 3.375 times as much.

What survives scaling and what does not

Angles survive untouched, which is the definition of similar working backwards: the enlarged plate has exactly the angles of the drawn one. Perimeters scale by k, the same as any length. Areas scale by k², volumes by k³. Anything measured in plain length units follows k, anything in squared units follows k², and keeping that sorted is most of the practical skill.

The factor also has a direction. Going from the small triangle to the large one, k = 1.5. Going the other way, k = 1/1.5 = 0.667. State which figure is the original whenever you quote a factor, because a bare 1.5 on a marked up drawing has burned people in both directions.

Mistakes that catch people out

• Dividing non corresponding sides. Correspondence follows angle order, or equivalently rank by length. A ratio from a mismatched pair is meaningless.
• Checking one pair and stopping. One agreeing ratio is a hypothesis. Three agreeing ratios are a verification. The third division is cheap insurance.
• Scaling areas by k. Areas grow by k squared. A 1.5 enlargement carries 2.25 times the area, and estimating material by 1.5 shorts the order by a third.
• Quoting the factor without direction. Small to large is 1.5, large to small is 0.667. Name the original or the number is ambiguous.
• Assuming similarity from a resemblance. Two triangles that look alike can still fail the ratio test. The agreement of all three ratios is the proof, not the eyeball.

Try these

Common questions

What is the scale factor between two similar triangles?

The ratio of any side of one triangle to the side it corresponds to in the other. All three corresponding pairs give the same ratio, and that shared value is the factor.

How do I know which sides correspond?

Sides opposite equal angles correspond. The quick method is to sort each triangle by side length and pair by rank, shortest with shortest through longest with longest.

How do areas change under a scale factor?

By the square of the factor. A factor of 1.5 multiplies every area by 2.25, and a factor of 3 multiplies areas by 9.

Can the scale factor be less than one?

Yes, it just means the second triangle is a reduction of the first. The factor from large to small is the reciprocal of the factor from small to large.

What if the three ratios disagree?

Then the triangles are not similar and no scale factor exists. A disagreement in one pair identifies which side has been distorted relative to a true enlargement.

Does the scale factor change the angles?

No. Similar figures have identical angles by definition. Scaling changes lengths, areas and volumes, never the angles.

Related lessons and calculators

• How to tell if two triangles are similar, the tests that establish similarity before any factor is quoted.
• How to tell if two triangles are congruent, the special case where the factor is exactly one.
• How to find the area from three sides, for computing the areas that the square law then relates.
• Triangle solver, to solve both triangles fully and watch the angles match.
• NCTM, the National Council of Teachers of Mathematics, on similarity and proportional reasoning.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.