1 : √3 : 2

30-60-90 Triangle Calculator

The 30° angle is pinned for you — enter just one side and get the other two using the exact 1 : √3 : 2 ratio.

Solve your 30-60-90 triangle

Angle α is fixed at 30° (so β is always 60°). Enter any one remaining value — a leg, the hypotenuse, the altitude, the area, or the perimeter — pre-loaded here with a hypotenuse of 10 mm.

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
30-60-90 triangle calculator diagram A 30-60-90 right triangle with hypotenuse 10 millimeters, short leg 5 millimeters, and long leg 8.66 millimeters. A B C a = 5 mm b = 8.66 mm c = 10 mm α = 30° β = 60°

Step-by-step solution

  1. Hypotenuse (given directly)
    c = c
    c = 10 mm
    = 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 makes 30-60-90 a "special" right triangle

A 30-60-90 triangle is a right triangle whose acute angles are exactly 30° and 60°. What makes it "special" is that its side lengths always fall in the exact ratio 1 : √3 : 2 — the side opposite the 30° angle (the short leg) is always half the hypotenuse, and the side opposite the 60° angle (the long leg) is always the short leg times √3. That means once you know just one side, the other two follow immediately without needing a calculator for the trigonometry — only a single multiplication or division by √3 or 2.

This calculator still runs the full solver underneath — pinning α to 30° and letting you supply any one of the six remaining quantities (either leg, the hypotenuse, the altitude to the hypotenuse, the area, or the perimeter) — so it works identically whether you're checking a textbook ratio problem or a real measurement that doesn't come out to a clean number.

Worked example: hypotenuse = 10

With the hypotenuse fixed at 10 mm and α = 30°, the short leg is a = c·sin(30°) = 10 × 0.5 = 5 mm, and the long leg is b = c·cos(30°) = 10 × 0.8660 ≈ 8.66 mm. Notice this matches the 1 : √3 : 2 ratio exactly: 5 : 8.66 : 10 simplifies to 1 : √3 : 2. From there, area = (1/2) × 5 × 8.66 ≈ 21.65 mm², perimeter ≈ 23.66 mm, the altitude to the hypotenuse ≈ 4.33 mm, the inradius ≈ 1.83 mm, and the circumradius is exactly half the hypotenuse, 5 mm.

When you'd use a 30-60-90 triangle

30-60-90 triangles show up constantly in geometry and trigonometry coursework, in equilateral-triangle and hexagon constructions, in roof-pitch and ramp-angle calculations that happen to use a 30° or 60° incline, and in any design that involves bisecting an equilateral triangle. If your triangle instead has a 45° angle, use the 45-45-90 triangle calculator, or switch between both types in the special right triangle calculator. For a general right triangle with any angle at all, use the full right triangle calculator.

30-60-90 triangle FAQ

What is the 30-60-90 rule?

The sides always sit in the exact ratio 1 : √3 : 2 — the side opposite the 30° angle (the short leg) to the side opposite the 60° angle (the long leg) to the hypotenuse. Because √3 is irrational, only the short-leg-to-hypotenuse and long-leg-to-hypotenuse ratios are ever "nice" round numbers when a triple like 3-4-5 is not also involved.

How can you tell if a triangle is a 30-60-90?

Check the angles first: if one angle is 90° and another is 30° or 60°, it is a 30-60-90 triangle. From the sides, check whether the hypotenuse is exactly twice the shortest side, or whether the longer leg is √3 (about 1.732) times the shortest.

Where does the 30-60-90 triangle come from?

Cut an equilateral triangle exactly in half through one vertex and the midpoint of the opposite side, and each resulting half is a 30-60-90 triangle: the original 60° angle stays 60°, the bisected 60° angle becomes 30°, and the cut creates a 90° angle. This is the cleanest way to derive the 1 : √3 : 2 ratio from scratch.

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).

See the full right triangle FAQ for more general questions.

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