This special right triangles calculator will help you to solve the chosen triangle in a blink of an eye. Select the triangle you need and type the given values - the remaining parameters will be calculated automatically. Special right triangles are right triangles for which simple formulas exist. That allows quick calculations, so you don't need to use the Pythagorean theorem or some advanced method. Scroll down to read more about special right triangles formulas and rules.

Special right triangles 30 60 90

Special right triangle 30° 60° 90° is one of the most popular right triangles. Its properties are so special because it's half of the equilateral triangle.

If you want to read more about that special shape, check our calculator dedicated to the 30° 60° 90° triangle.

Special right triangles 45 45 90

Another famous special right triangle is 45° 45° 90° triangle. It's the only possible right triangle that is also an isosceles triangle. Also, it's the shape created when we cut the square along the diagonal:

Curious about this triangle's properties? Have a look at our tool about the 45° 45° 90° triangle.

Other special right triangles

Many special right triangles exist, below you'll find the ones implemented in our tool:

Special right triangles formulas

If you are looking for the formulas for special right triangles, you are in the right place. Have a look at this neat table below and everything should be clear! In this table, you'll find the formulas for the relationship between special right triangle angles, legs, hypotenuse, area and perimeter:
Special right trianglea (shorter leg)b (longer leg)c (hypotenuse)AreaPerimeterAngle αAngle β
30° - 60° - 90° x x√32xx²√3/2x(3+√3)30°60°
45° - 45° - 90° xxx√2x²/2x(2+√2)45°45°
x - 2x x2xx√5x(3+√5)~26.5°~63.5°
x - 3xx3xx√103x²/2x(4+√10)~18.5°~71.5°
3x - 4x - 5x 3x4x5x6x²12x~37°~53°

Special right triangle rules

Special right triangles are the triangles that have some specific features which make the calculations easier. Of course, the most important special right triangle rule is that they need to have one right angle plus that extra feature. Generally, special right triangles may be divided into two groups:

  • Angle-based right triangles - for example 30°-60°-90° and 45°-45°-90° triangles
  • Side-based right triangles - figures that have side lengths governed by a specific rule, e.g.:
    • sides with integer lengths called Pythagorean triplets:

      3:4:5, 5:12:13, 8:15:17, 7:24:25, 9:40:41...

    • sides with integer lengths, but almost-isosceles:

      20:21:29, 119:120:169, 696:697:985...

    • right triangle, the sides of which are in a geometric progression (Kepler triangle). It's formed by three squares sides. Their areas are in geometric progression, according to the golden ratio.

There are many different rules and choices by which we can choose the triangle and call it special. In our special right triangles calculator, we implemented five chosen triangles: two angle-based and three side-based.

Hanna Pamuła, PhD student

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Special Right Triangles. Calculator | Formula | Rules