Similar Triangles Calculator: Scale Factor & Missing Sides (2026)
Calculate missing sides and scale factors of similar triangles. Enter known side lengths to find unknown sides and verify triangle similarity.
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Put these formulas into practice with our instant, step-by-step Similar Triangles Calculator.
Similar triangles are triangles that have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional. Understanding similar triangles is crucial for solving many geometry problems, from finding unknown distances to proving geometric theorems. Our free similar triangles calculator helps you find missing sides and verify similarity.
Key Takeaway
Similar triangles have proportional sides and equal angles. The scale factor k relates corresponding sides: a₁/a₂ = b₁/b₂ = c₁/c₂ = k. Area ratio = k².
What are Similar Triangles?
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The ratio of corresponding sides is called the scale factor (k).
Similarity Criteria
- AA (Angle-Angle): Two angles are equal
- SAS (Side-Angle-Side): Two sides are proportional and included angle is equal
- SSS (Side-Side-Side): All three sides are proportional
Key Properties
Perimeter ratio: P₁/P₂ = k
Area ratio: A₁/A₂ = k²
Worked Example
Two Similar Triangles with Scale Factor 2
Triangle 1 has sides 3, 4, 5. Triangle 2 is similar with scale factor k = 2.
| Property | Triangle 1 | Triangle 2 |
|---|---|---|
| Side a | 3 | 6 |
| Side b | 4 | 8 |
| Side c | 5 | 10 |
| Perimeter | 12 | 24 |
| Area | 6 | 24 |
Real-World Applications
- Surveying: Measuring inaccessible distances (height of buildings, width of rivers)
- Architecture: Scale models and blueprints
- Astronomy: Parallax measurements for stellar distances
- Photography: Lens calculations and field of view
- Map Making: Scale representations of terrain
Frequently Asked Questions
How do I find a missing side using similarity?
Set up a proportion using corresponding sides: a₁/a₂ = b₁/b₂ = c₁/c₂. Substitute the known values and solve for the unknown. For example, if 3/x = 4/8, then x = 6.
What is the ratio of areas of similar triangles?
If the scale factor is k, the ratio of areas is k². For example, if the scale factor is 3, the area ratio is 9:1. The larger triangle has 9 times the area of the smaller one.
How are similar triangles used in real life?
Similar triangles are used in surveying (measuring inaccessible distances), architecture (scale models), astronomy (parallax measurements), map making, photography, and shadow problems.
Try our similar triangles calculator to compute scale factors and missing sides instantly, or explore our right triangle calculator and equilateral triangle calculator for more triangle computations.