Special Right Triangles Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Sep 27, 2023

This special right triangles calculator will help you solve the chosen triangle's measurements in a blink of an eye. Select the triangle you need and type the given values – the remaining parameters will be calculated automatically.

The 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 triangle formulas and rules.

Special right triangles 30 60 90

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

Special right triangle: 30-60-90

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

Special right triangles 45 45 90

Another famous special right triangle is the 45°45\degree 45°45\degree 90°90\degree 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:

Special right triangle: 45-45-90Special right triangle: 45-45-90

Curious about this triangle's properties? Please look at our 45° 45° 90° triangle calculator.

Other special right triangles

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

Special right triangle: 3x-4x-5xSpecial right triangle: b=2aSpecial right triangle: b=3a

Special right triangles formulas

right triangle with sides a,b,c and angles α, β

If you are looking for the formulas for special right triangles, you are in the right place. Look at this neat table below; 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 triangle

aa (shorter leg)

bb (longer leg)

cc (hypotenuse)



Angle α\alpha

Angle β\beta

30°30\degree - 60°60\degree - 90°90\degree


x3\sqrt 3






45°45\degree - 45°45\degree - 90°90\degree



x2x\sqrt 2


x(2+2)x(2+\sqrt 2)



x2xx - 2x





x(3+5)x(3+\sqrt 5)



x3xx - 3x



x10x\sqrt {10}





3x3x - 4x4x - 5x5x








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°30\degree-60°60\degree-90°90\degree and 45°45\degree-45°45\degree-90°90\degree triangles.

  • Side-based right triangles – figures that have side lengths governed by a specific rule, e.g.:

    • Sides with integer lengths called Pythagorean triplets:

      3:4:53:4:5, 5:12:135:12:13, 8:15:178:15:17, 7:24:257:24:25, 9:40:419:40:41...

    • Sides with integer lengths but almost-isosceles:

      20:21:2920:21:29, 119:120:169119:120:169, 696:697:985696:697:985...

    • Right triangle, the sides of which are in a geometric progression (Kepler triangle). It's formed by three square sides. Their areas are in geometric progression, according to the golden ratio. For more on this special ratio, head to our golden ratio calculator.

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.

Special right triangle calculator – example

Let's have a look at the example: we want to find the length of the hypotenuse of a right triangle if the length of one leg is 55 inches and one angle is 45°45\degree.

  1. Choose the proper type of special right triangle. In our case, it's 45°45\degree-45°45\degree-90°90\degree triangle.

  2. Type in the given value. We know that the side is equal to 55 in, so we type that value in the a or b box – it doesn't matter where because it's an isosceles triangle.

  3. Wow! The special right triangle calculator solved the measurements of your triangle! Now we know that:

    • Second leg bb is equal to 5 in5\ \mathrm{in};
    • Hypotenuse cc is 7.07 in7.07\ \mathrm{in};
    • Perimeter equals 17.07 in17.07\ \mathrm{in}; and
    • Area of our special triangle is 12.5 in212.5\ \mathrm{in^2}.

Don't wait any longer. Try it yourself!


What are the formulas for a 45 45 90 triangle?

A 45° 45° 90° triangle has the following formulas, where x is the length of any of the equal sides:

  • Hypotenuse = x√2;
  • Area = x²/2; and
  • Perimeter = x(2+√2).

How do I solve a 30 60 90 special right triangle?

To solve a 30° 60° 90° special right triangle, follow these steps:

  1. Find the length of the shorter leg. We'll call this x.
  2. The longer leg will be equal to x√3.
  3. Its hypotenuse will be equal to 2x.
  4. The area is A = x²√3/2.
  5. Lastly, the perimeter is P = x(3 + √3).

What are the two special triangles in trigonometry?

30° 60° 90° triangles and 45° 45° 90° (or isosceles right triangle) are the two special triangles in trigonometry. While there are more than two different special right triangles, these are the fastest to recognize and the easiest to work with. An example of a non-angle-based special right triangle is a right triangle whose sides form a Pythagorean triple.

Is 3, 4, and 5 a Pythagorean triplet?

Yes. The integers a = 3, b = 4, and c = 5 form a Pythagorean triplet since a² + b² = c², and a triangle with sides abc is a right special triangle.

Hanna Pamuła, PhD
Special triangle
30° 60° 90°
Special right triangle: 30-60-90

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