Triangle Similarity Calculator

Created by Luis Hoyos
Reviewed by Arturo Barrantes
Last updated: Oct 21, 2022

If you're trying to prove triangle similarity, this triangle similarity calculator can greatly help.

With this triangle similarity geometry tool, you can:

  • Check triangle similarity for SSS, SAS, and ASA criteria; and
  • Find the missing side, area, perimeter, and scale factor between two similar triangles.

Read on to learn the three similarity statements for triangles used to prove triangle similarity.

Triangle similarity theorems

The general property of similar triangles is that two triangles are similar if and only if their three corresponding angles are congruent. Consequently, the three corresponding sides are equally proportional.

🔎 Two angles are congruent if they are identical to each other.

We don't have to verify all those conditions to get a triangle similarity proof, as there are three triangle similarity theorems that require less information.

Imagine two triangles: ABC and A'B'C', and see what those statements for triangle similarity say:

  • SSS (Side-Side-Side) criteria: this similar triangle property says there's similarity if all the corresponding sides are proportional. Mathematically:

    ABAB=BCBC=ACAC\frac{AB}{A'B'} =\frac{BC}{B'C'} = \frac{AC}{A'C'}

  • SAS (Side-Angle-Aide) criteria: If two pairs of sides are proportional and the angles between them are congruent, there's similarity:

    If ABAB=BCBC\frac{AB}{A'B'} =\frac{BC}{B'C'}, and ABC\angle{ABC} is congruent to ABC\angle{A'B'C'}, there's similarity.

  • AA (Angle-Angle) criteria: the AA triangle similarity criteria says if two pairs of corresponding angles are congruent, that's proof of triangle similarity. For example:

    If BAC\angle{BAC} is congruent to BAC\angle{B'A'C'} and ABC\angle{ABC} is congruent to ABC\angle{A'B'C'}, thanks to the angle sum property, we know that ACB\angle{ACB} is congruent to ACB\angle{A'C'B'} and the triangles are similar.

Other calculators for triangle similarity in geometry

Apart from this geometry triangle similarity tool, you can look at these other calculators:

FAQ

If the ratio of the angles of a triangle is 2 4 6, what is the measure of those angles?

In a 2 4 6 triangle, the measure of the angles are 30°, 60°, and 90°. Here's the explanation:

  1. As the ratio of the angles of the triangle is 2 4 6, we can say the three angles are:
    ϴ₁ = 2x, ϴ₂ = 4x and ϴ₃ = 6x.
  2. According to the angle sum property:
    2x + 4x + 6x = 180°
    12x = 180°
    x = 15°
  3. Finally:
    ϴ₁ = 30°, ϴ₂ = 60° and ϴ₃ = 90°.

How to check similarity in right triangles?

There are two ways to check right triangles similarity:

  • Acute angle similarity:
    1. Identify the corresponding angles between the two right triangles.
    2. Identify the two acute angles (those that measure less than 90°) for each triangle.
    3. If at least one of the acute angles is congruent with its corresponding acute angle of the other triangle, then the triangles are similar (thanks to the AA triangle similarity theorem).
  • Sides similarity:
    1. Identify the corresponding sides between the two triangles.
    2. If at least two sides are congruent to two corresponding sides of the other triangle, the triangles are similar.
Luis Hoyos
Type
Check similarity
Similarity criterion
Side-Side-Side (SSS)
△ABC
AB (a)
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BC (b)
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AC (c)
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△DEF
DE (d)
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EF (e)
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DF (f)
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