Similar Right Triangles Calculator

How to use this similar right triangles calculator

What this calculator does

This calculator completes and compares two right triangles that are intended to be similar. It uses the Pythagorean theorem to recover a missing side, checks that the hypotenuse is the longest side, and tests whether corresponding sides share one scale factor. It then reports both acute angles, perimeter, area, the perimeter ratio, and the area ratio. It does not prove that a physical drawing is accurately measured, and it does not handle oblique triangles whose included angle is not 90 degrees. The underlying rules are the standard properties of similar triangles: equal corresponding angles and proportional corresponding sides, as explained in the OpenStax guide to similar triangles and the Pythagorean theorem.

When to use it

Use the calculator to scale a right-triangle plan, check a reduced drawing against a full-size layout, recover missing dimensions from a proportional pair, or verify a classroom exercise. It is also useful when a diagonal brace, ramp, roof profile, or survey sketch is represented by two right triangles with corresponding sides.

How to calculate

1. The calculator opens with a complete 3 – 4 – 5 cm Triangle 1, a scale factor of 1.5, and a 4.5 – 6 – 7.5 cm Triangle 2. The live results and the example Excel workbook are ready immediately.
2. Choose the Length unit. Switching between centimeters, meters, inches, and feet converts all six side entries so the physical triangles remain unchanged.
3. Replace the demonstration values in Triangle 1 side a (leg), Triangle 1 side b (leg), and Triangle 1 side c (hypotenuse). Any two positive sides are enough; the third is calculated.
4. Enter the Scale factor (Triangle 2 ÷ Triangle 1), or clear it and provide at least two Triangle 2 sides. When both a scale factor and Triangle 2 sides are present, the calculator checks that they agree.
5. Read the primary similarity result, the Triangle 1 and Triangle 2 result cards, and the comparison table. Select Download Excel to export the current typed inputs and canonical results. Reset clears the demonstration and all calculated content; Excel remains unavailable until a complete valid set is entered again.

Input guide

Length unit is required and accepts cm, m, in, or ft. A realistic choice is cm for a classroom diagram. Unit changes convert side values; they do not change the scale factor. Each of the six side fields accepts a positive plain decimal in the selected unit, such as 3, 4.5, or 1,250.25. Commas must be conventional thousands separators, not decimal commas, and scientific notation is rejected. Triangle 1 side a (leg) and Triangle 1 side b (leg) are perpendicular legs; increasing either changes the hypotenuse, acute angles, perimeter, and area. Triangle 1 side c (hypotenuse) is the side opposite the right angle and must exceed either leg. Entering all three Triangle 1 sides is optional, but if all are supplied they must satisfy a² + b² = c² within numerical tolerance.

Scale factor (Triangle 2 ÷ Triangle 1) is optional only when at least two Triangle 2 sides are supplied. It must be greater than zero; 1 means equal size, a value above 1 enlarges Triangle 2, and a value between 0 and 1 reduces it. The fields Triangle 2 side a (leg), Triangle 2 side b (leg), and Triangle 2 side c (hypotenuse) use the same accepted format and correspond respectively to Triangle 1 sides a, b, and c. A common mistake is pairing a leg with the other triangle's hypotenuse; correspondence must stay a-to-a, b-to-b, and c-to-c.

Output guide

Similarity result states the validated scale factor and whether the corresponding measurements are proportional. Sides a, b, c reports the completed lengths in the active unit. Angles α, β gives the two acute angles in degrees; α is opposite side a and β is opposite side b, and they sum to 90°. Perimeter is a + b + c. Area is ab ÷ 2 and uses square units. Perimeter ratio equals the scale factor k, while Area ratio equals k². A scale factor below 1 produces a smaller Triangle 2; a factor above 1 produces a larger one. Zero or negative lengths are invalid rather than meaningful boundary results. The comparison table repeats each canonical result and labels whether a quantity is multiplied or stays equal.

Worked example

In the opening example, Triangle 1 has a = 3 cm and b = 4 cm. The missing-side identity gives c = √(3² + 4²) = √25 = 5 cm. Multiplying each side by k = 1.5 gives Triangle 2 sides 4.5 cm, 6 cm, and 7.5 cm. Triangle 1 area is 3 × 4 ÷ 2 = 6 cm². Triangle 2 area is 4.5 × 6 ÷ 2 = 13.5 cm², so the area ratio is 13.5 ÷ 6 = 2.25, exactly k². Both triangles have α = 36.87°, β = 53.13°, and a 90° angle.