Enter the three sides of triangle A and one side of triangle B to find all missing sides, the scale factor, area ratio, and perimeter ratio between similar triangles.
Two triangles are similar when their corresponding angles are equal and their corresponding sides are in proportion. The constant of proportionality is called the scale factor (k). If Triangle A has sides a₁, b₁, c₁ and Triangle B is similar with scale factor k, then B has sides ka₁, kb₁, kc₁. Perimeters scale linearly (ratio k) while areas scale quadratically (ratio k²).
Similarity is established through three criteria: AA (two pairs of equal angles), SSS (all three side ratios equal), and SAS (two sides in proportion with the included angle equal). This calculator focuses on the practical task: given three sides of one triangle and one side of the other, find all remaining sides and relationships.
Similar triangles appear constantly in real-world measurement. Shadow-length problems, aerial-photo scale calibration, map distance conversion, and architectural model-to-building scaling all rely on triangle similarity. In engineering, stress analysis on scaled models uses the square-of-scale-factor law for area-dependent quantities.
This calculator computes the scale factor from your inputs, derives all six sides, both perimeters, both areas, and displays the perimeter ratio and area ratio. A visual comparison bar chart lets you see the relative sizes at a glance, presets cover classic textbook examples, and the reference table lists common scale factors with their area multipliers.
This tool is valuable when you already know two triangles are similar and need the scaling consequences quickly. Instead of recalculating each missing side one by one and then separately handling perimeter and area changes, you can derive the full second triangle from a single matching side. That is useful for textbook similarity problems, scale drawings, model construction, and any workflow where one triangle serves as the reference shape for another.
Scale factor: k = (known side of B) / (corresponding side of A) Sides of B: a₂ = k·a₁, b₂ = k·b₁, c₂ = k·c₁ Perimeter ratio: P₂ / P₁ = k Area ratio: A₂ / A₁ = k² Area via Heron: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Result: k = 2, Triangle B = 10-24-26, Perimeter ratio = 2, Area ratio = 4
Triangle A has sides 5, 12, and 13. Because side c in Triangle B is 26, the scale factor is 26/13 = 2. Multiply each side of Triangle A by 2 to get Triangle B: 10, 24, and 26. Perimeters double, and areas multiply by 2² = 4.
Once two triangles are known to be similar, a single corresponding side is enough to determine the entire second triangle. That is the main idea behind this calculator. You enter all three sides of Triangle A, choose which side in Triangle B is known, and the scale factor fills in the rest. This mirrors the logic used in geometry classes: establish similarity first, then use proportional sides to recover unknown lengths.
Students often expect all measurements to scale in the same way, but area behaves differently. If the side lengths are multiplied by 2, the perimeter also doubles, yet the area becomes four times as large. If the scale factor is 0.5, the perimeter is halved while the area drops to one quarter. Seeing side lengths, perimeters, and Heron-based areas together makes that distinction much easier to internalize.
Similar-triangle calculations are central to map scales, blueprints, enlargement and reduction problems, and indirect measurement. In design work, a prototype and the final part may share the same shape with a fixed linear ratio. In classroom settings, these problems also connect several geometry skills at once: proportional reasoning, triangle classification, and area formulas. A calculator that keeps the correspondences explicit helps avoid one of the most common mistakes, which is pairing the wrong side in Triangle B with the wrong side in Triangle A.
Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. The shape is the same; only the size may differ.
Divide any side of the second triangle by the corresponding side of the first: k = side₂ / side₁. Use matching sides (shortest to shortest, longest to longest).
The area ratio equals the scale factor squared. If k = 3, the larger triangle has 9 times the area of the smaller one.
AA: two pairs of equal angles. SSS: all three pairs of sides in the same ratio. SAS: two pairs of sides in the same ratio with the included angle equal.
Yes — when the scale factor k = 1, the similar triangles are the same size and are therefore congruent. Use this as a practical reminder before finalizing the result.
Shadow-length problems, map scales, architectural models, aerial photo calibration, and stress analysis on scaled engineering prototypes all use similar-triangle relationships. Keep this note short and outcome-focused for reuse.