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Right Triangle Similarity Calculator

Comparing right triangles requires checking whether their corresponding sides maintain a constant ratio. This calculator verifies similarity by testing the proportionality of side lengths across two right triangles. Enter the dimensions of both triangles—either two legs or all three sides—to instantly confirm whether they share the same shape, and discover the scale factor relating them.

Last updated:

Creators Anna Szczepanek, PhD Mathematics
7,234 people find this calculator helpful

Understanding Right Triangle Similarity

Two right triangles are similar when they have identical angle measures or when their corresponding sides are proportional. Since all right triangles contain a 90° angle, similarity hinges on whether the other two angles match.

The key indicator of similarity is the scale factor: a constant multiplier showing how much larger or smaller one triangle is compared to the other. If you scale every side of triangle A by the same number and get triangle B, the triangles are similar.

For example, a 3–4–5 triangle and a 6–8–10 triangle are similar because each side of the larger triangle is exactly twice the corresponding side of the smaller one. The scale factor is 2.

Similarity Verification Through Proportionality

To confirm similarity, calculate the scale factor for each pair of corresponding sides. If all ratios are identical, the triangles are similar.

First, verify each triangle satisfies the Pythagorean theorem:

c² = a² + b²

Then compare the ratios of corresponding sides:

scale_factor = side₂ / side₁

The triangles are similar if this ratio is constant across all three side pairs.

  • a, b — The two legs (perpendicular sides) of a right triangle
  • c — The hypotenuse (longest side opposite the right angle)
  • scale_factor — The constant ratio between corresponding sides of the two triangles

How to Use This Calculator

The calculator accepts input in two modes depending on what information you have about the second triangle.

  • Two legs known: Enter the lengths of both legs, and the calculator will compute the hypotenuse using the Pythagorean theorem, then test similarity against the first triangle.
  • All three sides known: Provide all three side lengths for complete verification. This mode also allows you to input a scale factor to test whether that specific relationship holds.

After calculation, the tool displays whether the triangles are similar and reveals the exact scale factor. A reflection may also be detected if one triangle is a mirror image of the other.

Common Pitfalls When Comparing Right Triangles

Avoid these mistakes when checking triangle similarity.

  1. Confusing similarity with congruence — Similar triangles have the same shape but not necessarily the same size. Congruent triangles are identical in both shape and size (scale factor = 1). Two right triangles can be similar without being congruent.
  2. Calculating the hypotenuse incorrectly — When only two legs are given, always use c² = a² + b², not approximations. A small rounding error in the hypotenuse cascades into incorrect scale factors and false similarity verdicts.
  3. Ignoring the scale factor consistency — The scale factor must be identical for all three side pairs. If two sides scale by 2 and the third by 2.1, the triangles are not similar. Consistent proportionality across all sides is non-negotiable.
  4. Forgetting to verify the Pythagorean theorem — Before comparing, ensure both triangles actually satisfy c² = a² + b². Invalid side lengths (those that violate the theorem) may give misleading similarity results.

The 3–4–5 and 6–8–10 Example

A classic pair demonstrating similarity: the 3–4–5 and 6–8–10 triangles.

Both satisfy the Pythagorean theorem:

  • 3–4–5: 3² + 4² = 9 + 16 = 25 = 5²
  • 6–8–10: 6² + 8² = 36 + 64 = 100 = 10²

Comparing side ratios reveals a consistent scale factor of 2:

  • 6 ÷ 3 = 2
  • 8 ÷ 4 = 2
  • 10 ÷ 5 = 2

Since all ratios match, these triangles are similar. The 6–8–10 triangle is simply an enlarged version of the 3–4–5 triangle, scaled uniformly by a factor of 2.

Frequently Asked Questions

What is the difference between similar and congruent right triangles?

Similar triangles share identical angles and proportional sides but may differ in size. Congruent triangles are identical in every dimension—they have the same shape and size, making their scale factor exactly 1. Two congruent right triangles are always similar, but similar triangles are congruent only when the scale factor equals one.

How do I find the scale factor between two right triangles?

Divide any side length from the second triangle by the corresponding side from the first. For example, if triangle 1 has a leg of 4 and triangle 2 has the corresponding leg of 8, the scale factor is 8 ÷ 4 = 2. Calculate this ratio for all three pairs of sides. If all three ratios are identical, the triangles are similar, and that common ratio is your scale factor.

Can two right triangles be similar if only two sides match the proportions?

No. For definitive similarity, all three pairs of corresponding sides must have the same proportional relationship. If two pairs scale by a factor of 2 and the third by 1.9, the triangles are not similar. The proportionality must be constant across every side pair, which is why the Pythagorean theorem helps verify this—if c² = a² + b² and the scale factors for a and b match, then c's scale factor must also match.

What if the sides of my second triangle don't form a right triangle?

If a² + b² ≠ c² for the second triangle, it is not a valid right triangle, and similarity testing becomes meaningless. The calculator will flag this issue. Always verify that both triangles satisfy the Pythagorean theorem before proceeding with similarity checks.

How does the calculator detect reflections?

A reflection occurs when one triangle is a mirror image of the other. Both triangles are still similar because they share the same angles and proportional sides—just oriented differently. The calculator identifies this by checking whether corresponding sides maintain the scale factor even when the triangle orientation differs, noting whether sides map in the expected or reversed order.

Are all 3–4–5 triangles similar to each other?

Yes. Any triangle with sides in the 3–4–5 ratio is similar to every other 3–4–5 triangle, regardless of actual size. A 6–8–10 triangle, a 9–12–15 triangle, and a 30–40–50 triangle are all similar because their side ratios remain constant. Similarity depends only on proportions, not absolute dimensions.

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