Special Right Triangles Calculator

Created by Hanna Pamuła, PhD candidate
Reviewed by Bogna Szyk
Last updated: May 10, 2022

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.

Special right triangle: 30-60-90

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:

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

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 triangle: 3x-4x-5x
Special 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. 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.

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 the one leg is 5 inches and one angle is 45°.

  1. Choose the proper type of special right triangle. In our case, it's 45°-45°-90° triangle.
  2. Type in the given value. We know that the side is equal to 5 in, so we type that value in a or b box - it doesn't matter where because it's an isosceles triangle.
  3. Wow! The special right triangle calculator solved your triangle! Now we know that:
  • Second leg b is equal to 5 in
  • Hypotenuse c is 7.07 in
  • Perimeter equals 17.07 in
  • Area of our special triangle is 12.5 in².

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 non-angle-based special right triangles 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 candidate
Special triangle
30° 60° 90°
Special right triangle: 30-60-90

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