What Is a 30-60-90 Triangle?
A 30-60-90 triangle is a right triangle with angles of 30°, 60°, and 90°. Unlike the isosceles 45-45-90 triangle, its two legs are unequal: the side opposite 30° (the short leg) is always exactly half the hypotenuse, and the side opposite 60° (the long leg) is the short leg multiplied by √3.
The fixed angle proportions mean the sides always follow the same ratio regardless of size:
a : b : c = 1 : √3 : 2 (short leg : long leg : hypotenuse)
Formulas
For a 30-60-90 triangle with short leg a:
| Measurement | Formula | Example (a = 5 cm) |
|---|---|---|
| Long leg (b) | b = a√3 | ≈ 8.660 cm |
| Hypotenuse (c) | c = 2a | 10 cm |
| Area (A) | A = a²√3 / 2 | ≈ 21.651 cm² |
| Perimeter (P) | P = a(3 + √3) | ≈ 23.660 cm |
Reverse formulas
If you know a value other than the short leg:
- From long leg: a = b / √3
- From hypotenuse: a = c / 2
- From area: a = √(2A / √3)
- From perimeter: a = P / (3 + √3)
How to Use This Calculator
Enter any one known value — short leg, long leg, hypotenuse, area, or perimeter — and the remaining four update instantly. The formula box shows the active relationship. Use the unit selector to switch between metric (mm, cm, m, km) and imperial (in, ft, yd, mi) units.
Connection to the Equilateral Triangle
The 30-60-90 triangle is half of an equilateral triangle. Draw the altitude of an equilateral triangle with side s and it splits into two mirror-image 30-60-90 triangles where:
- Hypotenuse c = s (the original side)
- Short leg a = s / 2 (half the base)
- Long leg b = s√3 / 2 (the altitude)
This relationship makes the 30-60-90 triangle essential in regular hexagon and equilateral triangle geometry.
Trigonometric Values at 30° and 60°
The 30-60-90 triangle gives exact trigonometric values that appear throughout mathematics:
| Function | 30° | 60° |
|---|---|---|
| sin | 1/2 | √3/2 ≈ 0.866 |
| cos | √3/2 ≈ 0.866 | 1/2 |
| tan | 1/√3 ≈ 0.577 | √3 ≈ 1.732 |
Worked Examples
Example 1 — From the short leg
A ramp rises at 30° over a horizontal run of 6 m. Find the ramp length and height.
The horizontal run is the long leg: b = 6 m, so a = b / √3 = 6 / 1.732 ≈ 3.464 m (height), and c = 2a ≈ 6.928 m (ramp length).
Example 2 — From the hypotenuse
An equilateral triangle has sides of 8 cm. What is its altitude?
The altitude is the long leg of the 30-60-90 triangle with hypotenuse = 8 cm:
- a = 8 / 2 = 4 cm
- b = 4√3 ≈ 6.928 cm
Example 3 — From the area
A 30-60-90 triangle has an area of 30 cm². Find its dimensions.
- a = √(2 × 30 / √3) = √(60 / 1.732) = √34.641 ≈ 5.886 cm
- b = 5.886 × √3 ≈ 10.196 cm
- c = 2 × 5.886 ≈ 11.772 cm