Triangle Similarity Calculator
Enter matching sides as a, b, c. Side a is opposite angle A. Side b is opposite angle B. Side c is opposite angle C.
Similarity Graph
The graph draws both triangles when valid side lengths are available. Otherwise, it can show angle differences after calculation.
Example Data Table
| Case | Triangle One | Triangle Two | Expected Result | Reason |
|---|---|---|---|---|
| Scaled 3-4-5 | a=3, b=4, c=5 | a=6, b=8, c=10 | Similar | All side ratios equal 2. |
| AA proof | A=50°, B=60°, C=70° | A=50°, B=60°, C=70° | Similar | Two or more angles match. |
| Failed side ratio | a=5, b=7, c=8 | a=10, b=14, c=19 | Not similar | The third side ratio differs. |
| SAS proof | b=6, c=8, A=45° | b=12, c=16, A=45° | Similar | Two side ratios and included angle match. |
Formula Used
SSS similarity: Triangles are similar when a₂/a₁ = b₂/b₁ = c₂/c₁.
AA similarity: Triangles are similar when two matching angles are equal. The third angle must also match because each triangle totals 180°.
SAS similarity: Triangles are similar when two matching side ratios are equal and the included angle is equal.
Scale factor: k = matching side of Triangle Two / matching side of Triangle One.
Law of cosines for derived angles: A = cos⁻¹((b² + c² - a²) / 2bc).
How to Use This Calculator
- Enter side lengths for both triangles when known.
- Enter angles when side lengths are missing or rounded.
- Use matching labels for corresponding sides and angles.
- Adjust tolerance for measured or estimated values.
- Press the submit button to view the result above the form.
- Review the proof method, graph, scale factor, and notes.
- Use the CSV or PDF button to save your result.
Triangle Similarity Guide
What Similarity Means
Triangle similarity is a key idea in geometry. It tells you when two triangles share the same shape. Their sizes may differ, yet their angles match. Their matching sides also keep one constant ratio. This calculator checks that relationship with several proof paths.
Proof Rules
You can test sides, angles, or both. The side test compares the three side ratios. This is the SSS similarity rule. The angle test compares matching angle sets. This follows the AA rule, because two equal angles force the third angle to match. The mixed test uses two sides and the included angle. This is the SAS rule.
Tolerance and Scale
A tolerance option is included for real measurements. Drawings, scans, and classroom data often include small rounding errors. A strict comparison may reject a correct pair. A sensible tolerance gives a fair result. You can lower it for exact textbook values. You can raise it for field measurements.
Understanding the Output
The result explains which rule passed. It also shows a scale factor. The scale factor tells how much Triangle Two is enlarged or reduced from Triangle One. A factor of 2 means every matching side is twice as long. A factor below 1 means the second triangle is smaller.
Visual Review
The graph helps you see the shapes. When side lengths are valid, both triangles are drawn on a shared plane. This makes the comparison easier. The table also lists example data. You can use it to understand expected input order.
Common Uses
This tool is useful for homework, design checks, surveying, map scaling, and model work. It supports quick learning because the proof is visible. It does not only say yes or no. It shows the reason behind the decision.
Input Tips
For best results, enter sides in matching order. Use side a opposite angle A, side b opposite angle B, and side c opposite angle C. If you only know angles, enter at least two angles for each triangle. If you know all sides, leave angles blank. The calculator can derive angles from valid side lengths.
It also encourages careful labeling. Correct labels prevent false failures. When labels are uncertain, sort sides mentally first. Then compare the smallest, middle, and largest sides during review work.
FAQs
1. What makes two triangles similar?
Two triangles are similar when their matching angles are equal and their matching sides stay in one constant ratio. They have the same shape, but they may have different sizes.
2. Can similar triangles have different areas?
Yes. Similar triangles can have different areas. If the side scale factor is k, the area scale factor is k squared. A triangle with double side lengths has four times the area.
3. What is the AA similarity rule?
The AA rule says two triangles are similar when two matching angles are equal. The third angle must also be equal because every triangle has a 180 degree angle sum.
4. What is the SSS similarity rule?
The SSS rule checks all three matching side ratios. If each ratio is equal, the triangles are similar. This calculator compares those ratios within your selected tolerance.
5. What is the SAS similarity rule?
The SAS rule uses two matching side ratios and the included angle between them. If the ratios match and the included angle matches, the triangles are similar.
6. Why does tolerance matter?
Tolerance handles rounded or measured values. Real drawings and field measurements are rarely perfect. A small tolerance helps avoid rejecting triangles that are practically similar.
7. What does the scale factor mean?
The scale factor tells how much Triangle Two changes from Triangle One. A factor above 1 means Triangle Two is larger. A factor below 1 means it is smaller.
8. Do I need to enter every angle?
No. Enter at least two angles for each triangle when using angle comparison. If you enter valid side lengths, the calculator can derive the missing angles automatically.