Calculator Input
Example Data Table
Use these sample values to test side ratio matching and missing side solving.
| Case | Triangle 1 sides | Triangle 2 sides | Expected result |
|---|---|---|---|
| Classic match | a=3, b=4, c=5 | a′=6, b′=8, c′=10 | Similar, k=2 |
| Missing side | a=5, b=7, c=9 | a′=10, b′=14, c′ blank | c′=18 |
| Angle proof | A=40°, B=60° | A′=40°, B′=60° | Similar by AA |
Formula Used
For similar triangles, corresponding sides keep one constant scale factor:
k = a′ / a = b′ / b = c′ / c
Missing matching side formulas are:
a′ = k × a, b′ = k × b, c′ = k × c, and a = a′ / k.
Perimeter ratio equals k. Area ratio equals k². The angle sum rule is A + B + C = 180°.
Similarity checks use AA, SSS, and SAS. AA checks matching angles. SSS checks all side ratios. SAS checks two side ratios and their included angle.
How to Use This Calculator
- Enter matching sides in the same positions: a with a′, b with b′, and c with c′.
- Add any known angles. Leave unknown fields blank.
- Choose a tolerance for rounded or measured values.
- Click the calculate button.
- Review the scale factor, solved lengths, similarity tests, chart, and working steps.
- Use CSV or PDF download buttons to save your result.
Similar Triangles Guide
Similar triangles are triangles with the same shape.
Their sizes may be different. Matching angles are equal. Matching sides stay in one constant ratio. This calculator helps you use that idea with less guesswork.
Why Similar Triangles Matter
Similar triangles appear in algebra, geometry, surveying, drawing, maps, construction, and physics. A small model can predict a larger object. A shadow can estimate a height. A diagram can find a missing side. The method is powerful because it turns shape comparison into clean ratios.
What The Calculator Solves
Enter any known matching side pairs. Add optional angles when you have them. The tool finds the scale factor from the first triangle to the second triangle. It can solve missing matching sides when at least one valid side ratio exists. It also checks similarity with SSS, SAS, and AA rules. You also get perimeter ratio, area ratio, and clear working steps.
How To Read The Results
The scale factor is written as k. If k is 2, the second triangle is twice as large in every matching length. If k is 0.5, it is half as large. Perimeters follow the same ratio. Areas follow the square of the ratio. So a scale factor of 3 gives an area ratio of 9.
Good Input Habits
Use the same unit for all side lengths. Match side a with a′, b with b′, and c with c′. Do not mix unrelated sides. Angles should also match by position. If you know two angles in a triangle, the third angle can be found from 180 degrees. The calculator can fill that value for you.
Practical Use
Students can check homework steps. Teachers can prepare examples. Designers can resize plans. Builders can compare slopes and shapes before making final measurements. The chart gives a quick visual check, while the CSV and PDF buttons help save the solution for later review.
Use Tolerance Carefully
Real measurements often have small errors. A classroom problem may need exact ratios. A field measurement may allow a small tolerance. Raise the tolerance only when rounded or measured data is expected. Keep it low for exact symbolic work. It supports online review.
FAQs
1. What are similar triangles?
Similar triangles have the same shape. Their matching angles are equal, and their matching sides are proportional. They may have different sizes.
2. What does the scale factor mean?
The scale factor tells how much the second triangle is enlarged or reduced compared with the first triangle. It equals a′ divided by a.
3. Can I leave fields blank?
Yes. Leave unknown sides or angles blank. The calculator solves values when enough matching side or angle data is available.
4. Which similarity tests are used?
The calculator checks AA, SSS, and SAS. It compares matching angles, side ratios, and included angles using your chosen tolerance.
5. Why is area ratio different?
Length changes by the scale factor. Area changes by the square of that factor. If k is 4, the area ratio is 16.
6. What tolerance should I use?
Use a low tolerance for exact homework problems. Use a larger tolerance for rounded measurements, field data, or construction estimates.
7. Do side labels matter?
Yes. Corresponding sides must be placed in matching fields. Side a must match a′, b must match b′, and c must match c′.
8. Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable solution summary.