Similar Triangles Calculator
Enter the three sides of triangle A and one known side of similar triangle B (any consistent unit). The calculator returns the scale factor, both missing sides, and how the perimeter and area scale.
Example: with Triangle A — side a 6 · Triangle A — side b 8 · Triangle A — side c 10 · Triangle B — side matching a 9 → Scale factor (k): 1.5× (each side of B = 1.5 × matching side of A).
- Triangle B — side matching b12 units
- Triangle B — side matching c15 units
- Perimeters (A → B)24 → 36 units (scales by k)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Similar triangles: equal angles, sides in one fixed ratio k; perimeter scales by k, area by k².
Scale factor first, everything else follows
Two triangles are similar when their angles match, which forces every pair of corresponding sides into one fixed ratio — the scale factor k. Find k from the single pair you know (k = a₂/a₁, here 9/6 = 1.5) and the missing sides are one multiplication away: b₂ = k·b₁ and c₂ = k·c₁. The perimeter scales by the same k, since it is just a sum of sides.
The whole method stands on matching the right sides. Follow the letters of the similarity statement (△ABC ~ △DEF pairs AB with DE), or match shortest to shortest and longest to longest.
Area is the trap
Area does not scale by k — it scales by k². Doubling every side quadruples the area; the default 1.5× enlargement multiplies area by 2.25 (the 6-8-10 triangle's area of 24 becomes 54). The calculator computes both areas with Heron's formula so you can see the square law in actual numbers, and the same logic extends to volumes of similar solids, which scale by k³.
How it’s calculated
Scale factor k = a₂/a₁ from the matched pair. Missing sides: b₂ = k·b₁, c₂ = k·c₁. Perimeter scales by k; area by k². Actual areas come from Heron's formula, √(s(s − a)(s − b)(s − c)) with s the half-perimeter. Rounded to 3 decimals.
Assumes the triangles really are similar and that the sides you paired correspond; mismatched pairing produces a wrong but plausible-looking factor.
What scales by how much (k = 1.5 example)
| Quantity | Scales by | 6-8-10 triangle × 1.5 |
|---|---|---|
| Side lengths | k | 8 → 12 |
| Perimeter | k | 24 → 36 |
| Angles | unchanged | 53.13° → 53.13° |
| Area | k² | 24 → 54 |
| Volume (similar solids) | k³ | × 3.375 |
Standard similarity relations; the example scales this page's default 6-8-10 right triangle by k = 1.5.
Common mistakes
- Pairing non-corresponding sides — follow the similarity statement's letter order, or match shortest with shortest and longest with longest.
- Applying the scale factor to area: area scales by k², not k.
- Inverting the factor — decide which triangle is the original and which is the copy before dividing.
- Assuming similar means congruent; similar triangles share shape, and only k = 1 makes them the same size.
Frequently asked questions
How do I find the scale factor of similar triangles?
Divide a side of the second triangle by its corresponding side in the first: k = a₂/a₁. Every other pair of corresponding sides then obeys the same ratio, which is what makes the missing sides solvable.
How do areas of similar triangles compare?
By the square of the scale factor: area₂ = k² × area₁. A triangle scaled 1.5× in every side has 2.25× the area — a fact this page demonstrates with Heron's formula on your actual numbers.
How do I know which sides correspond?
Use the order of the similarity statement (△ABC ~ △DEF matches A↔D, B↔E, C↔F), or rank the sides: shortest pairs with shortest, longest with longest. Corresponding sides always sit opposite equal angles.
Do the angles change when a triangle is scaled?
No — equal angles are the definition of similarity. Scaling stretches lengths by k but leaves every angle exactly as it was.