Similar Right Triangles Calculator

Enter two sides of the first right triangle and one side of the second to find the scale factor and all missing sides.

Triangle 1 (Reference)

Enter at least two sides. The third will be calculated via the Pythagorean theorem.

Triangle 2 (Similar)

Enter one known side. Select which side it corresponds to.

Result

Scale Factor (k)
--
Triangle 1: Leg a --
Triangle 1: Leg b --
Triangle 1: Hypotenuse c --
Triangle 2: Leg a' --
Triangle 2: Leg b' --
Triangle 2: Hypotenuse c' --
Angle A (opposite leg a) --
Angle B (opposite leg b) --
Angle C (right angle) 90°
Area of Triangle 1 --
Area of Triangle 2 --

Step-by-Step Solution

What Are Similar Right Triangles?

Two right triangles are similar if they have the same acute angles. Since both already have a 90-degree angle, they only need one more matching angle to be similar. Similar right triangles have proportional sides, meaning the ratio of corresponding sides is constant (the scale factor).

Key Concepts

Scale Factor

The ratio between corresponding sides of two similar triangles.

k = side2 / side1

Pythagorean Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

c² = a² + b²

Area Ratio

The ratio of areas of similar triangles equals the square of the scale factor.

Area2/Area1 = k²

How to Find Missing Sides

  1. Complete Triangle 1 using the Pythagorean theorem if needed.
  2. Identify which side of Triangle 2 is known and its corresponding side in Triangle 1.
  3. Calculate the scale factor: k = known side of T2 / corresponding side of T1.
  4. Multiply each side of Triangle 1 by k to get all sides of Triangle 2.
  5. Verify: the computed sides should satisfy the Pythagorean theorem.

Altitude on Hypotenuse Theorem

When you draw an altitude from the right angle to the hypotenuse in a right triangle, it creates two smaller right triangles that are both similar to the original triangle and to each other. This is a powerful tool in geometry proofs and problem-solving.

Applications

  • Architecture: scaling building designs while maintaining proportions.
  • Trigonometry: establishing sine, cosine, and tangent ratios.
  • Surveying: using shadow measurements to calculate heights.
  • Engineering: model-to-prototype scaling in mechanical design.
  • Photography: understanding perspective and similar triangles in optics.