How this instrument works
When two triangles are similar, every corresponding pair of sides shares the identical scale factor. Given a full reference triangle (all three sides) and just ONE known side of a second, similar triangle, this calculator first finds that shared scale factor by dividing the known corresponding sides, then applies it to the reference triangle's other two sides to recover the second triangle's missing measurements: b₂ = b₁ × (a₂⁄a₁) and c₂ = c₁ × (a₂⁄a₁).
This carries the calculation one step further than this site's companion Triangle Scale Factor page, which stops at reporting the ratio itself. Here, that ratio is put straight to work, solving the ENTIRE second triangle from a single known correspondence — the practical version of the same underlying idea, useful whenever a full similar triangle needs reconstructing rather than just its scale confirmed.
This only works when the two triangles are already known or assumed to be genuinely similar — matching angles, proportional sides. If that assumption doesn't actually hold, applying a single ratio to the other two sides won't correctly describe the real second triangle at all, since a shared scale factor is only valid between shapes that are truly proportional to begin with.
- Enter the first triangle's three side lengths into the Triangle 1 fields.
- Enter the second, similar triangle's one known corresponding side into the Triangle 2: corresponding side a field.
- Read Triangle 2: side b (solved) and Triangle 2: side c (solved): the sheet finds the scale factor and applies it to the other two sides.
Worked example — triangle (3,4,5), corresponding side 6
Triangle 1 has sides 3, 4, and 5. A similar triangle 2 has a corresponding side of 6 in place of the 3, a scale factor of 2. The remaining two sides scale by that identical factor: b₂ = 4×2 = 8 and c₂ = 5×2 = 10 — the full second triangle recovered from just one known corresponding side.
Triangle 1 with sides 5, 12, and 13, and a corresponding side 2 of exactly 10 (scale factor 2), solves to b₂=24 and c₂=26. And a corresponding side 2 equal to triangle 1's original 3 gives a scale factor of exactly 1 — the two triangles are congruent, and the solved sides simply match triangle 1's own originals.
Questions
How do you solve a similar triangle from just one known side?
First find the scale factor by dividing the known corresponding sides, then multiply the reference triangle's other two sides by that identical factor — since similar triangles share one uniform scale factor across every pair of corresponding sides.
How is this different from the Triangle Scale Factor page?
That page reports only the scale factor itself. This page uses that same factor to go further, solving the second triangle's two remaining missing sides directly, rather than stopping at the ratio alone.
What if the triangles aren't actually similar?
This calculation assumes genuine similarity (matching angles, proportional sides) between the two triangles. If that assumption doesn't hold, applying a single scale factor to the other two sides won't correctly describe the real second triangle.
What does a scale factor of exactly 1 mean here?
The two triangles are congruent, not merely similar — identical in both shape and size. The 'solved' sides simply come out matching the first triangle's own originals exactly.
Can the second triangle be smaller than the first?
Yes — if the known corresponding side of triangle 2 is smaller than triangle 1's matching side, the scale factor comes out between 0 and 1, and the solved sides correctly shrink rather than grow.