Similar Triangles Calculator

Determine similarity between two triangles, compute the scale factor, find unknown side lengths, and visualize proportional relationships. Based on AA, SSS, SAS similarity theorems.

Triangle A (Reference)
Standard 3-4-5 right triangle (example). Must satisfy triangle inequality.
Triangle B (Similar)
? 3-4-5 → Scale 2 → 6-8-10
? Right 5-12-13 → Scale 1.5 → 7.5-18-19.5
? Equilateral side 4 → Scale 0.75 → side 3
✅ Check similarity: 6-8-10 & 9-12-15 (k=1.5)
❌ Non-similar demo: 4-5-6 vs 8-9-12

Understanding Similar Triangles: Core Theorems & Applications

Two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional. The constant of proportionality is the scale factor (k). Similarity preserves shape but not size, forming the basis of scale drawings, shadow measurements, and indirect measurement — a technique used since Thales of Miletus (circa 600 BCE).

If ΔABC ~ ΔA'B'C', then: ∠A = ∠A', ∠B = ∠B', ∠C = ∠C' and a/a' = b/b' = c/c' = k (or a' = k·a).

Three Principal Similarity Theorems

  • AA (Angle-Angle): If two angles of one triangle equal two angles of another, the triangles are similar. The most powerful and common criterion.
  • SSS (Side-Side-Side): If all three side ratios are equal (a/a' = b/b' = c/c'), the triangles are similar.
  • SAS (Side-Angle-Side): If two sides are proportional and the included angle is congruent, the triangles are similar.
Real‑world Application: Measuring Heights with Shadows

Eratosthenes measured Earth's circumference using similar triangles. In modern times, surveyors use a 2-meter pole and its shadow to calculate the height of a tree or building. If the pole casts a 1.5m shadow and a tree casts a 12m shadow, the tree height = (pole height × tree shadow) / pole shadow = (2 × 12)/1.5 = 16 meters. Our calculator automates proportional reasoning.

How Our Similar Triangles Calculator Works

The tool checks triangle validity (triangle inequality) for both triangles, then computes side ratios (a₂/a₁, b₂/b₁, c₂/c₁). If all three ratios are equal within a tolerance of 1e-6, the triangles are declared similar, and the scale factor is returned. If side lengths for triangle B are incomplete but a scale factor k is provided, the calculator generates missing sides of triangle B directly, ensuring perfect similarity. The system also validates numeric stability and provides clear alerts for degenerate or non‑similar cases.

Step-by-step guide

  1. Enter three side lengths for Triangle A (positive numbers satisfying triangle inequality).
  2. Either fill sides for Triangle B OR enter a scale factor k (e.g., 2.0) and click "Apply k".
  3. Click "Analyze Similarity & Compute" to get scale factor, status, and precise side ratios.
  4. Visual representation displays both triangles (auto‑scaled for comparison).

Scale Factor Properties & Euler‑like Invariance

The scale factor k relates perimeters, altitudes, medians, and angle bisectors: all linear measures scale by k, area scales by k². The concept links directly to homothety and geometric transformations. Unlike the orthocenter in non‑similar triangles, similar triangles share proportional altitude lengths and corresponding segment ratios are invariant — a foundational concept in fractal geometry and scaling laws.

Triangle Pair Side set A Side set B Scale factor k Similar?
3-4-5 & 6-8-10 3,4,5 6,8,10 2.0 Yes (SSS)
5-12-13 & 7.5-18-19.5 5,12,13 7.5,18,19.5 1.5 Yes
Right isosceles (1,1,√2) 1,1,1.414 2,2,2.828 2.0 Yes
4-5-6 vs 8-9-12 4,5,6 8,9,12 2.0,1.8,2.0 No (ratios differ)

History & Mathematical Significance

Thales used similar triangles to calculate the height of the Great Pyramid. Euclid’s Elements (Book VI) formalized proportional segments. The concept of similarity is central to trigonometry, coordinate geometry, and modern computer vision (scale-invariant feature transform). Our calculator implements exact rational comparisons when possible and floating-point tolerance for real‑world inputs.

Frequently Asked Questions

Yes, if you have AA (two angles) you can still infer similarity. This calculator focuses on side lengths (SSS) and scale factor, but you can compute missing sides if you know k. For angle‑only inputs, we recommend using complementary trigonometric tools.

The calculator will compute the third side using the scale factor derived from the two known sides if possible, provided the ratios match. Otherwise, it suggests providing all three sides or explicit k for full accuracy.

Yes, sides are paired in order: side a (opposite vertex A), side b (opposite B), side c (opposite C). Ensure consistent correspondence; the tool assumes a ↔ a', b ↔ b', c ↔ c'. For arbitrary ordering you can rearrange values manually.

The canvas uses exact side lengths to build triangles using law of cosines. The drawn triangles reflect true shape ratios and are scaled independently for optimal fit; they are geometrically faithful to the side lengths entered.
References: Euclid’s Elements, Book VI; Weisstein, Eric W. "Similar Triangles." MathWorld; National Council of Teachers of Mathematics (NCTM) standards for similarity.