Special Right Triangles Calculator (2024)

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Special right triangles 30 60 90Special right triangles 45 45 90Other special right trianglesSpecial right triangles formulasSpecial right triangle rulesSpecial right triangle calculator – exampleFAQs

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\degree30° 60°60\degree60° 90°90\degree90° is one of the most popular right triangles. Its properties are unique because it's half of the equilateral triangle.

Special Right Triangles Calculator (1)

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\degree45° 45°45\degree45° 90°90\degree90° 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 Triangles Calculator (2)Special Right Triangles Calculator (3)

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 Triangles Calculator (4)Special Right Triangles Calculator (5)Special Right Triangles Calculator (6)

Special right triangles formulas

Special Right Triangles Calculator (7)

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

aaa (shorter leg)

bbb (longer leg)

ccc (hypotenuse)

Area

Perimeter

Angle α\alphaα

Angle β\betaβ

30°30\degree30° - 60°60\degree60° - 90°90\degree90°

xxx

x3x\sqrt 3x3

2x2x2x

x23/2x^2\sqrt{3}/2x23/2

x(3+3)x(3+\sqrt3)x(3+3)

30°30\degree30°

60°60\degree60°

45°45\degree45° - 45°45\degree45° - 90°90\degree90°

xxx

xxx

x2x\sqrt 2x2

x2/2x^2/2x2/2

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

45°45\degree45°

45°45\degree45°

xxx - 2x2x2x

xxx

2x2x2x

x5x\sqrt5x5

x2x^2x2

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

26.5°\sim26.5\degree26.5°

63.5°\sim63.5\degree63.5°

xxx - 3x3x3x

xxx

3x3x3x

x10x\sqrt {10}x10

3x2/23x^2/23x2/2

x(4+10)x(4+\sqrt{10})x(4+10)

18.5°\sim18.5\degree18.5°

71.5°\sim71.5\degree71.5°

3x3x3x - 4x4x4x - 5x5x5x

3x3x3x

4x4x4x

5x5x5x

6x26x^26x2

12x12x12x

37°\sim37\degree37°

53°\sim53\degree53°

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\degree30°-60°60\degree60°-90°90\degree90° and 45°45\degree45°-45°45\degree45°-90°90\degree90° triangles.

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

    • Sides with integer lengths called Pythagorean triplets:

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

    • Sides with integer lengths but almost-isosceles:

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

    • 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 555 inches and one angle is 45°45\degree45°.

  1. Choose the proper type of special right triangle. In our case, it's 45°45\degree45°-45°45\degree45°-90°90\degree90° triangle.

  2. Type in the given value. We know that the side is equal to 555 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 bbb is equal to 5in5\ \mathrm{in}5in;
    • Hypotenuse ccc is 7.07in7.07\ \mathrm{in}7.07in;
    • Perimeter equals 17.07in17.07\ \mathrm{in}17.07in; and
    • Area of our special triangle is 12.5in212.5\ \mathrm{in^2}12.5in2.

Don't wait any longer. Try it yourself!

FAQs

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.

Special Right Triangles Calculator (2024)

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