Similar Triangles Calculator
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.
Check Similarity in Right Triangles — Side Ratio & Angle Comparison
Enter the sides of two right triangles to check if they are similar. Compare side ratios, scale factor, angle measurements, area ratio, and perimeter ratio.
Triangle 1
Triangle 2
Common Pythagorean Triples
| Triple | Angles | Family |
|---|---|---|
| 3 – 4 – 5 | 36.87° – 53.13° – 90° | Primitive |
| 5 – 12 – 13 | 22.62° – 67.38° – 90° | Primitive |
| 8 – 15 – 17 | 28.07° – 61.93° – 90° | Primitive |
| 7 – 24 – 25 | 16.26° – 73.74° – 90° | Primitive |
| 9 – 40 – 41 | 12.68° – 77.32° – 90° | Primitive |
| 6 – 8 – 10 | 36.87° – 53.13° – 90° | 2 × (3-4-5) |
| 10 – 24 – 26 | 22.62° – 67.38° – 90° | 2 × (5-12-13) |
| 20 – 21 – 29 | 43.60° – 46.40° – 90° | Primitive |
Planning notes, formulas, and examples
About the Check Similarity in Right Triangles — Side Ratio & Angle Comparison
Two triangles are similar if and only if their corresponding angles are equal (or, equivalently, their corresponding side lengths are proportional). For right triangles, checking similarity is especially simple: both triangles already share a 90° angle, so you only need one more pair of equal acute angles — or equivalently, one pair of proportional legs — to establish similarity.
The scale factor k is the ratio of corresponding sides. If triangle 2 has sides that are each k times those of triangle 1, then their perimeters also differ by factor k, while their areas differ by k². This makes similarity a powerful tool in real-world applications: architects use scale models, mapmakers rely on proportional distances, and engineers apply similarity to stress analysis.
Pythagorean triples are integer-sided right triangles. The family 3-4-5 includes every multiple (6-8-10, 9-12-15, etc.), all of which are similar to each other. But 3-4-5 and 5-12-13 are not similar — their acute angles differ. Recognising which triples belong to the same family is a common geometry exercise.
This calculator accepts the sides of two right triangles, verifies each is indeed right-angled, sorts and compares their side ratios, and declares whether the triangles are similar. It reports the scale factor, all acute angles, area and perimeter ratios, and shows comparison bars and a reference table of common Pythagorean triples.
When This Page Helps
Use this when you need to prove two right triangles are or are not similar without manually sorting side ratios and recomputing the acute angles. It is useful for scale drawings, trig checks, and textbook exercises because the verdict, scale factor, and area ratio all come from the same pair of triangles.
How to Use the Inputs
- Choose input mode: enter 3 sides each, or 2 legs each (hypotenuse computed automatically).
- Enter the side lengths for Triangle 1.
- Enter the side lengths for Triangle 2.
- Or click a preset to load a known pair.
- View the similarity verdict, scale factor, angles, area ratio, and perimeter ratio.
- Compare visually in the bars and the side-by-side table.
Formula used
SSS Similarity: Two triangles are similar iff a₁/a₂ = b₁/b₂ = c₁/c₂ (sorted sides).
Scale factor: k = corresponding side ratio.
Area ratio = k².
Perimeter ratio = k.
Right triangle check: a² + b² = c².Example Calculation
Result: For mode=three-sides, a1=3, b1=4, the tool returns the solved check similarity in right triangles — side ratio & angle comparison outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in check similarity in right triangles — side ratio & angle comparison formulas and reports derived values, checks, and classifications automatically.
Tips & Best Practices
- For right triangles, matching just one acute angle is sufficient to prove similarity (AA criterion).
- All multiples of a Pythagorean triple (e.g., 6-8-10, 9-12-15) are similar to the primitive triple (3-4-5).
- The area ratio is the square of the scale factor — doubling the sides quadruples the area.
- If the side ratios don't match, the triangles are NOT similar, no matter how close they look.
- A 45-45-90 triangle is only similar to other 45-45-90 triangles, and a 30-60-90 only to other 30-60-90s.
When To Use This Calculator
Enter the sides of two right triangles to check if they are similar. Compare side ratios, scale factor, angle measurements, area ratio, and perimeter ratio. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
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Frequently Asked Questions
Sort each triangle's sides from smallest to largest. Compute the ratios of corresponding sides. If all three ratios are equal, the triangles are similar.
The scale factor k is the common ratio of corresponding sides. If k = 2, triangle 2 is twice as large as triangle 1.
No. Two right triangles are similar only if they share the same acute angles. A 3-4-5 and a 5-12-13 are both right triangles but not similar.
Area ratio = k². If the sides are scaled by k, the area scales by k². For example, if k = 3, the larger triangle has 9× the area.
A Pythagorean triple is a set of three positive integers (a, b, c) with a² + b² = c². The simplest is (3, 4, 5). All integer multiples of a primitive triple form a family of similar right triangles.
This calculator is designed for right triangles and verifies the right-angle property. For general triangles, use a general similarity checker that compares all three angle pairs.
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