30-60-90 • 45-45-90

Special Right Triangle Calculator

Pick 30-60-90 or 45-45-90, enter one side, and get the other sides, area, and perimeter instantly, with every step shown.

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Solve a special right triangle

Choose the triangle type, then enter any one value: a leg, the hypotenuse, the altitude, the area, or the perimeter. It starts with a 30-60-90 set square, the drafting triangle from a geometry set, whose longest edge is 10 in.

Triangle type

Enter one known value

Fill in one field below. The angles are fixed, so everything else is solved automatically.

Angle α = 30° (fixed)
Angle mode

Results

30-60-90 right triangle
Leg a
5 in
Leg b
8.6603 in
Hypotenuse c
10 in
Angle α
30°
Angle β
60°
Altitude h
4.3301 in
Area
21.6506 in²
Perimeter
23.6603 in
Inradius
1.8301 in
Circumradius
5 in
Special right triangle calculator diagram: 30-60-90 triangle A 30-60-90 set square: short leg 5 inches, long leg 8.66 inches, and hypotenuse 10 inches. A B C a = 5 in b ≈ 8.6603 in c = 10 in α = 30° β = 60°

Step-by-step solution

The working keeps extra decimal places so you can check it by hand. Answers are rounded to your chosen decimal places.

  1. Find leg a
    a = c·sin(α)
    a = 10·sin(30°)
    = 5 in
  2. Find leg b
    b = c·cos(α)
    b = 10·cos(30°)
    ≈ 8.6603 in
  3. Find angle β
    β = 90° − α
    β = 90° − 30°
    = 60°
  4. Compute the area
    Area = (1/2)·a·b
    Area = (1/2)·5·8.660254
    ≈ 21.6506 in²
  5. Compute the perimeter
    P = a + b + c
    P = 5 + 8.660254 + 10
    ≈ 23.6603 in
  6. Compute the altitude to the hypotenuse
    h = (a·b) / c
    h = (5·8.660254) / 10
    ≈ 4.3301 in
  7. Inradius (radius of the inscribed circle)
    r = (a + b − c) / 2
    r = (5 + 8.660254 − 10) / 2
    ≈ 1.8301 in
  8. Circumradius (radius of the circumscribed circle)
    R = c / 2
    R = 10 / 2
    = 5 in

What are special right triangles?

Special right triangles are right triangles whose angles make the side lengths follow a fixed, exact ratio. There are two of them: the 45-45-90 triangle and the 30-60-90 triangle. Because the ratio never changes, you only need one measurement to solve the whole triangle, and you can often do it without trigonometry. That is why they turn up so often in geometry classes, standardized tests, and trigonometry tables.

Triangle Angles Side ratio Area (from shortest side s)
45-45-90 45°, 45°, 90° 1 : 1 : √2 s² / 2
30-60-90 30°, 60°, 90° 1 : √3 : 2 (√3 / 2) × s²

How to solve a 45-45-90 triangle

A 45-45-90 triangle is an isosceles right triangle: its two legs are equal, and the hypotenuse is a leg times √2. If you know a leg s, the hypotenuse is s√2. If you know the hypotenuse c, each leg is c / √2. For example, the 45° set square from a geometry set, with legs of 8 in, has a hypotenuse of 8√2 ≈ 11.314 in and an area of 8² / 2 = 32 in². For more detail and examples, use the isosceles right triangle calculator.

How to solve a 30-60-90 triangle

In a 30-60-90 triangle, the short leg sits opposite the 30° angle, the long leg sits opposite the 60° angle, and the hypotenuse is always exactly twice the short leg. From a short leg s, the long leg is s√3 and the hypotenuse is 2s. The 30-60-90 set square pre-loaded above has a 10 in hypotenuse, so its short leg is 10 / 2 = 5 in, its long leg is 5√3 ≈ 8.660 in, and its area is ½ × 5 × 5√3 ≈ 21.65 in². The dedicated 30-60-90 triangle calculator covers this case in more depth.

Where special right triangles come from

Cut a square along its diagonal and you get two 45-45-90 triangles. Cut an equilateral triangle in half from a corner to the middle of the opposite side and you get two 30-60-90 triangles. Those two shapes explain both ratios: the diagonal of a unit square is √2, and half of an equilateral triangle with side 2 has a short leg of 1 and a height of √3. In practice they show up in roof pitches, stair stringers, tile layouts, and anywhere a square or hexagon is involved.

If your triangle has any other angle, it isn't a special right triangle. Use the general right triangle calculator, which solves any right triangle from two known values, including its area and angles.

Special right triangle FAQ

What are the two special right triangles?

The 45-45-90 triangle (an isosceles right triangle with sides 1 : 1 : √2) and the 30-60-90 triangle (sides 1 : √3 : 2). They are called special because their side ratios are exact, so one side is enough to find the other two.

How many values do I need for a special right triangle?

Just one. The angles are already known, so a single side, the area, the perimeter, or the altitude fixes the size of the triangle. A general right triangle needs two values.

How do I know which special right triangle I have?

Look at the angles or the legs. If the two legs are equal, or both acute angles are 45°, it is a 45-45-90 triangle. If one angle is 30° or 60°, or the hypotenuse is exactly twice the shortest side, it is a 30-60-90 triangle.

How is a 30-60-90 triangle different from a 45-45-90 triangle?

A 30-60-90 triangle is scalene — all three sides have different lengths, in ratio 1 : √3 : 2. A 45-45-90 triangle is isosceles — the two legs are equal, in ratio 1 : 1 : √2. Both are called "special" right triangles because their exact ratios can be derived without a calculator, but they come from different constructions (halving an equilateral triangle versus halving a square).

Are there other special right triangles?

Most textbooks name just these two, the angle-based special right triangles: their angles are in simple ratios (1 : 1 : 2 and 1 : 2 : 3), and their sides follow ratios built from √2 and √3. Some books also call right triangles with whole-number sides, such as 3-4-5 and 5-12-13, side-based special right triangles; those are Pythagorean triples, and the calculator labels them too. Any other right triangle needs the general calculator and two known values.

See the full right triangle FAQ for more general questions.

Cite this page

To cite this calculator in schoolwork or a report, you can use this reference (APA style):

Right Triangle Calculators. (2026, October 10). Special right triangle calculator. https://righttrianglecalculators.com/special-right-triangle-calculator

References

The formulas and definitions on this page match these standard references.

  1. Weisstein, Eric W. “Isosceles Right Triangle.” MathWorld, Wolfram Research.
  2. Weisstein, Eric W. “30-60-90 Triangle.” MathWorld, Wolfram Research.
  3. Weisstein, Eric W. “Right Triangle.” MathWorld, Wolfram Research.

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