Similar Triangle Side Calculator – Solve for Missing Sides
Two triangles are similar when they have the same shape at different sizes: every pair of corresponding angles is equal, and every pair of corresponding sides shares one constant ratio. If triangle ABC is similar to triangle DEF, then d / a = e / b = f / c = k, where k is the scale factor. This calculator assumes the similarity is given and does the work you actually need at that point — finding the sides you do not have.
How the Missing Side Is Found
A single complete corresponding pair fixes the whole relationship. Enter a = 6 and d = 9 and the calculator computes k = 9 / 6 = 1.5. Every other blank then falls out by cross-multiplication: with b = 8, the matching side is e = b × k = 12. Solving in the other direction is the same identity read backwards, b = e / k. You never need to set up the proportion by hand, but the step-by-step block shows exactly how it was set up so the working can be copied into homework or a lesson.
Entering a Scale Factor or a Ratio Instead
Sometimes the ratio is what you are given — a map legend, a drawing scale, a model kit. Type it straight into the scale factor field as a decimal such as 2.4, or express it as a ratio like 2 : 3, read as Triangle 1 : Triangle 2. With a scale factor supplied, one complete triangle is enough to generate the other outright.
Why the Area Ratio Is k², Not k
This is the point where marks are most often lost. Perimeter is a sum of lengths, so it scales by plain k. Area is a product of two lengths, so it scales by k². A triangle enlarged by a factor of 1.5 has a perimeter 1.5 times bigger but an area 2.25 times bigger. The results panel prints length, perimeter, and area ratios as three bars precisely so that the gap between k and k² is visible rather than merely stated.
s = (a + b + c) / 2, Area = √(s(s−a)(s−b)(s−c)), and the shared angles come from the law of cosines, cos C = (a² + b² − c²) / 2ab. Similar triangles share those angles exactly, which is why one set is reported for both.Indirect Measurement With Shadows
The shadow mode solves the classic indirect measurement problem. A pole of known height and its shadow form a right triangle; the tree you cannot climb and its shadow form a similar one, because the sun's rays strike both at the same angle. A 1.8 m pole casting a 2.4 m shadow beside a tree casting a 14 m shadow gives 1.8 × (14 / 2.4) = 10.5 m. The tool also reports the sun's elevation angle as a sanity check on the measurement.
The Side-Splitter (Thales) Theorem
Not every similar pair sits side by side. When a line is drawn parallel to one side of a triangle, it cuts the other two sides proportionally and creates a smaller similar triangle nested inside the original. This is the basic proportionality theorem, also called the side-splitter or Thales theorem: AD / DB = AE / EC. Given AD = 4, DB = 6, AE = 5, the calculator returns EC = (6 × 5) / 4 = 7.5, along with the full sides AB and AC and the scale factor of the inner triangle against the whole.
Validation That Catches Bad Correspondences
Supplying more than one corresponding pair does not give the tool two competing ratios — it gives it a consistency check. If one pair implies k = 1.50 and another implies k = 1.62, the triangles are not similar under that vertex mapping, and the calculator says so instead of quietly picking the first one. Both completed side sets are also tested against the triangle inequality — the sum of any two sides must exceed the third — so an impossible triangle is flagged rather than reported as a result.
a is matched with d, b with e, and c with f. If your problem pairs the vertices differently, reorder the sides as you enter them, or use the Swap Triangles button to invert which triangle is treated as the original.Units and Precision
Each triangle has its own unit selector across millimetres, centimetres, metres, kilometres, inches, feet, yards, and miles. Mixed-unit input is converted to a common base internally before any ratio is formed, because the scale factor is dimensionless — a note states which conversion was applied. Rounding happens only for display; the calculation chain keeps full floating-point precision throughout, and you can set the displayed decimal places anywhere from 0 to 10.