Free Triangle Similarity Calculator

About the Triangle Similarity Calculator

The Triangle Similarity Calculator helps you quickly verify whether two triangles are similar using the SSS, SAS, and AA similarity criteria. Beyond checking similarity, it also computes missing side lengths, areas, perimeters, and the scale factor between the two triangles. Instead of performing manual proportion checks, you can enter the known values and get immediate results.

What Makes Triangles Similar?

Two triangles are considered similar when all three pairs of corresponding angles are equal (congruent) and the lengths of their corresponding sides are in the same ratio. This ratio, known as the scale factor, remains constant across all three side pairs. In essence, one triangle is an exact scaled version of the other, with no shape distortion.

The Three Key Similarity Criteria

Rather than verifying every angle and side, geometry provides three efficient similarity statements:

• SSS (Side‑Side‑Side) – If the ratios of all three corresponding side pairs are identical, the triangles are similar. For triangles $\triangle{ABC}$ and $\triangle{A'B'C'}$:

$$\frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{AC}{A'C'}$$

• SAS (Side‑Angle‑Side) – If two pairs of sides are in proportion and the included angles are congruent, similarity holds. For example:

$$\frac{AB}{A'B'} = \frac{BC}{B'C'} \quad \text{and} \quad \angle{ABC} = \angle{A'B'C'}$$

• AA (Angle‑Angle) – If any two angles of one triangle equal two angles of the other, the third angles must also be equal (since the sum of interior angles is always $180^\circ$). This alone is sufficient to prove similarity. For instance:

$$\angle{BAC} = \angle{B'A'C'} \quad \text{and} \quad \angle{ABC} = \angle{A'B'C'}$$

Checking Similarity and Finding Missing Values

Using this similar triangles calculator, you can test similarity under any of the three criteria. Provide the side lengths and/or angles of both triangles, and the tool will indicate whether they are similar. If similarity is confirmed, the scale factor is calculated automatically. From the scale factor, any unknown side can be found: multiply the known side by the scale factor (or its reciprocal) to obtain the missing length.

The calculator also outputs the areas and perimeters of the two triangles. Because area scales with the square of the scale factor and perimeter scales linearly, these results become instantly available once similarity is established.

Why Use This Tool?

Whether you are a student solving geometry problems, a teacher preparing examples, or a professional needing a quick geometric check, the Triangle Similarity Calculator saves time and reduces errors. It provides immediate feedback on SSS, SAS, and AA criteria and helps you find missing side lengths without manual proportion work.