About

Two right triangles are similar when their corresponding acute angles are equal. Since one angle is already fixed at 90°, only one additional angle match is needed (the AA criterion reduces to a single-angle check). Getting the similarity assessment wrong propagates errors in scaled engineering drawings, surveying computations, and trigonometric substitutions. This calculator compares two right triangles using either their leg/hypotenuse measurements or their acute angles, determines the applicable similarity criterion, and computes the scale factor k. It assumes Euclidean geometry and validates that inputs form valid right triangles via the Pythagorean constraint a² + b² = c² before comparison.

Formulas

For a right triangle with legs a, b and hypotenuse c, the Pythagorean theorem must hold:

c = √a² + b²

Acute angles are derived from sides using inverse trigonometric functions:

α = arctan(ab)

β = 90° − α

Two right triangles ▵₁ and ▵₂ are similar if their sorted acute angle sets match. For the SSS criterion, the scale factor is computed as:

k = a₁a₂

Similarity holds when all ratios of sorted corresponding sides are equal within tolerance ε:

|a₁a₂ − b₁b₂| < ε

Where a = shorter leg, b = longer leg, c = hypotenuse, α = angle opposite shorter leg, β = angle opposite longer leg, k = scale factor (ratio of corresponding sides), ε = comparison tolerance (default 0.01).

Reference Data

Criterion	Condition for Right Triangles	Minimum Data Needed	Notes
AA (Angle-Angle)	One acute angle of ▵1 = one acute angle of ▵2	1 acute angle per triangle	Strongest for right triangles; one angle suffices since the right angle is shared
SSS (Side-Side-Side)	a₁a₂ = b₁b₂ = c₁c₂	All 3 sides per triangle	Ratios must be equal within tolerance
SAS (Side-Angle-Side)	Two sides proportional ∧ included angle equal	2 sides + included angle	In right triangles, the right angle is always included between legs
HL (Hypotenuse-Leg)	Hypotenuse and one leg proportional	Hypotenuse + 1 leg	Special case of SSS for right triangles (congruence variant)
Scale Factor	k = side of ▵1corresponding side of ▵2	At least 1 pair of corresponding sides	k = 1 means congruent
Common Right Triangle	3-4-5	-	Angles ≈ 36.87° and 53.13°
Common Right Triangle	5-12-13	-	Angles ≈ 22.62° and 67.38°
Common Right Triangle	8-15-17	-	Angles ≈ 28.07° and 61.93°
Common Right Triangle	7-24-25	-	Angles ≈ 16.26° and 73.74°
Common Right Triangle	9-40-41	-	Angles ≈ 12.68° and 77.32°
Isosceles Right Triangle	1-1-√2	-	Angles: 45° and 45°
30-60-90 Triangle	1-√3-2	-	Angles: 30° and 60°
Area Ratio	k²	Scale factor k	Areas scale as the square of the linear scale factor
Perimeter Ratio	k	Scale factor k	Perimeters scale linearly with k
Tolerance	0.01 (1%)	-	Default ratio comparison tolerance for floating-point inputs

Frequently Asked Questions

Every right triangle already shares a 90° angle. The AA (Angle-Angle) similarity postulate requires two angle matches. Since the right angle is guaranteed, matching just one acute angle (e.g., α₁ = α₂) automatically satisfies the criterion. The third angle is determined because α + β = 90°.

The calculator uses a configurable tolerance ε of 0.01 (1%). Two ratios are considered equal if their absolute difference falls below this threshold. This accommodates real-world measurements where legs might be 3.001 instead of 3.000. For critical engineering applications, tighten the tolerance in the advanced settings.

If three sides are entered and a² + b² ≠ c² within tolerance, the calculator flags the input as an invalid right triangle. It will not force-fit the data. You can enter only two sides (two legs, or one leg and the hypotenuse) and the calculator derives the third side automatically.

No. If the linear scale factor is k, the area ratio is k². For example, if ▵₁ has legs 3 and 4, and ▵₂ has legs 6 and 8, then k = 2, but the area ratio is 4. The calculator displays both the scale factor and the area ratio.

Yes. Consider a 3-4-5 triangle and a 1-√24-5 triangle. Both have hypotenuse 5, but their acute angles differ (36.87° vs 11.54°). Equal hypotenuse alone does not imply similarity. At least one pair of corresponding sides must be proportional, or one pair of acute angles must match.

Sides of each triangle are sorted in ascending order (shortest leg, longest leg, hypotenuse). The calculator then pairs them positionally: smallest with smallest, middle with middle, largest with largest. This ensures the correct geometric correspondence regardless of the input order.