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.

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 triangle whose short leg is 5 mm.

Triangle type

Enter any two known values

Fill in exactly two fields below (a pinned value counts as one). Everything else is solved automatically.

Angle α = 30° (fixed)
Angle mode

Results

30-60-90 right triangle
Leg a
5 mm
Leg b
8.6603 mm
Hypotenuse c
10 mm
Angle α
30°
Angle β
60°
Altitude h
4.3301 mm
Area
21.6506 mm²
Perimeter
23.6603 mm
Inradius
1.8301 mm
Circumradius
5 mm
Special right triangle calculator diagram: 30-60-90 triangle A 30-60-90 special right triangle with short leg 5 millimeters, long leg 8.66 millimeters, and hypotenuse 10 millimeters. A B C a = 5 mm b = 8.66 mm c = 10 mm α = 30° β = 60°

Step-by-step solution

  1. Find the hypotenuse
    c = a / sin(α)
    c = 5 mm / sin(30°)
    = 10 mm
  2. Find the acute angles
    α = atan(a / b), β = 90° − α
    atan(5 / 8.6603)
    = α = 30°, β = 60°
  3. Compute the area
    Area = (1/2)·a·b
    Area = (1/2)·5·8.6603
    = 21.6506 mm²
  4. Compute the perimeter
    P = a + b + c
    P = 5 + 8.6603 + 10
    = 23.6603 mm
  5. Compute the altitude to the hypotenuse
    h = (a·b) / c
    h = (5·8.6603) / 10
    = 4.3301 mm
  6. Inradius (radius of the inscribed circle)
    r = (a + b − c) / 2
    r = (5 + 8.6603 − 10) / 2
    = 1.8301 mm
  7. Circumradius (radius of the circumscribed circle)
    R = c / 2
    R = 10 / 2
    = 5 mm

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, legs of 7 give a hypotenuse of 7√2 ≈ 9.899 and an area of 7² / 2 = 24.5. 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. With the short leg of 5 pre-loaded above, the long leg is 5√3 ≈ 8.660, the hypotenuse is 10, and the area is ½ × 5 × 8.660 ≈ 21.65. 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.

See the full right triangle FAQ for more general questions.

Related calculators

Right triangle calculator

Find the area, sides, and angles of any right triangle from any 2 known values.

30-60-90 triangle calculator

Solve a 30-60-90 triangle from a single known side.

Isosceles right triangle calculator

Solve a 45-45-90 isosceles right triangle from a single known side.

Pythagorean theorem calculator

Find the missing side of a right triangle, or check if 3 sides form one.

Right triangle formulas

Every right-triangle formula in one reference sheet.