Solve similar triangles by entering known sides to find missing lengths, scale factor, area ratio, perimeter ratio, and all triangle properties.
Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. This fundamental relationship means that knowing just a few measurements in each triangle lets you deduce all the rest. Our Similar Triangles Solver asks you to enter the three sides of the reference triangle and at least one side of the second triangle. It then calculates the scale factor k—the constant ratio between corresponding sides—and uses it to find every missing length. Beyond basic side calculations, the tool computes the perimeter ratio (equal to k), the area ratio (equal to k²), and the full geometric properties of both triangles: area via Heron's formula, all three angles via the law of cosines, the semi-perimeter, circumradius, and inradius. A visual bar chart compares corresponding sides at a glance, and a reference table lists the similarity criteria (AA, SAS, SSS) along with the core proportion formulas. Eight presets cover classic textbook configurations from simple 3-4-5 scalings to more complex ratios, so you can explore similarity relationships without manual input. This calculator is essential for geometry students, architects working with scaled drawings, and engineers using model-to-prototype conversions.
Similar-triangle work is simple in theory but easy to derail in practice because one wrong correspondence breaks every later proportion. This calculator keeps the reference triangle fixed, applies the scale factor consistently, and then extends that ratio to perimeter, area, angles, circumradius, and inradius automatically. That makes it much easier to confirm whether your setup is correct before you commit to a long solution.
It also helps when the goal is interpretation rather than arithmetic. In scale drawings, map work, model building, and geometry proofs, you often need to understand what changes linearly and what changes quadratically. Seeing the scale factor alongside the area ratio makes that relationship immediate.
Scale factor k = a₂/a₁ = b₂/b₁ = c₂/c₁. Perimeter ratio = k. Area ratio = k². Area (Heron) = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2.
Result: 6 (corresponding to a₁ = 3)
Triangle 1 has sides 3, 4, 5. Triangle 2 has side a₂ = 6 (corresponding to a₁ = 3). Scale factor k = 6/3 = 2. So b₂ = 4 × 2 = 8, c₂ = 5 × 2 = 10. Perimeter ratio = 2. Area ratio = 4. Area₁ = 6, Area₂ = 24.
The core idea in similar triangles is not just that the side lengths are proportional, but that the correct sides must be paired in the correct order. The side opposite the largest angle in one triangle must correspond to the side opposite the largest angle in the other, and the same logic applies to the remaining sides. When students get inconsistent results, the issue is often not arithmetic but mismatched correspondence. This solver makes that structure explicit by anchoring everything to triangle 1 and a chosen side in triangle 2.
One of the most useful insights from similar triangles is that linear measurements grow by the scale factor $k$, but areas grow by $k^2$. If a model is enlarged by a factor of 3, each side becomes 3 times longer, the perimeter becomes 3 times larger, and the area becomes 9 times larger. That distinction appears in architecture, map reading, image resizing, and fabrication. The calculator surfaces both ratios together so you can see the geometric consequence of scaling instead of treating it as a memorized rule.
Similarity shows up far beyond textbook proportion problems. Surveyors estimate inaccessible distances using nested triangles, artists preserve perspective through scaled shapes, and engineers translate small prototypes into full-size parts. In formal geometry, similar triangles are also a proof tool: once angle equality or side proportionality is established, you can infer new side lengths, parallel lines, or area relationships. A calculator like this is useful not because it replaces reasoning, but because it lets you test whether the proportions you set up really behave the way similarity predicts.
Two triangles are similar if their corresponding angles are equal (AA criterion) or if all corresponding sides are proportional (SSS similarity) or two sides are proportional with the included angle equal (SAS similarity). Use this as a practical reminder before finalizing the result.
The scale factor k is the constant ratio between any pair of corresponding sides: k = a₂/a₁ = b₂/b₁ = c₂/c₁. Keep this note short and outcome-focused for reuse.
The area ratio equals the square of the scale factor. If k = 2, then the larger triangle has 4 times the area of the smaller one.
Yes. Corresponding angles of similar triangles are always equal, regardless of the size difference.
Absolutely. Architecture, engineering, and cartography all rely on similar-triangle principles for scale conversions.
The calculator checks the triangle inequality (any two sides must sum to more than the third). If the check fails, no results are shown.