The 30 60 90 Triangle Calculator computes the full set of side lengths for a right triangle with angles 30°, 60°, and 90°. Enter one known side (in any common unit), and it returns the missing sides using the exact 1 : √3 : 2 ratio.
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle where the angles are fixed at 30°, 60°, and 90°. Because the angles are known, the side lengths always follow the same constant ratio.
In any 30-60-90 triangle:
- The side opposite 30° is the short leg.
- The side opposite 60° is the long leg.
- The side opposite 90° is the hypotenuse.
The 1 : √3 : 2 side ratio (core formulas)
Let the short leg (opposite 30°) be x. Then the long leg and hypotenuse are:
| Triangle side | Exact value | Meaning |
|---|---|---|
| Short leg (30° side) | x | Opposite 30° |
| Long leg (60° side) | x√3 | Opposite 60° |
| Hypotenuse (90° side) | 2x | Opposite 90° |
This is the reason the calculator is fast: once you know one side, the other sides are determined exactly.
If you know the short leg (x)
- Long leg = x√3
- Hypotenuse = 2x
If you know the long leg (x√3)
If the long leg is given, set x√3 = L. Then x = L/√3, so:
- Short leg = L/√3
- Hypotenuse = 2L/√3
If you know the hypotenuse (2x)
If the hypotenuse is given as H = 2x, then x = H/2, so:
- Short leg = H/2
- Long leg = (H/2)√3
Unit handling: inches, feet, meters, and more
Side lengths can be measured in many units (mm, cm, m, km, in, ft, yd, mi). The calculator converts your input into a consistent base unit internally, performs the triangle math, then converts results back to the unit you selected.
That means you can enter, for example, 12 inches and still get outputs in centimeters (or vice versa) without doing manual conversions.
How to use the 30 60 90 Triangle Calculator
- Select which side you know: short leg (30° side), long leg (60° side), or hypotenuse.
- Enter the numeric value.
- Choose the unit for your input.
- Pick the unit you want for the results.
- Click Calculate to get all three side lengths.
If you enter an invalid value (like 0 or a negative number), the calculator flags the field and asks you to correct it.
Practical examples (real-world use)
Example 1: Building a ramp or support
Suppose you need a right triangular support where one angle is 30° and you measure a short leg as 0.5 m. A 30-60-90 triangle lets you compute the exact long leg and hypotenuse without approximation.
Using the ratio:
- Long leg = 0.5√3 m
- Hypotenuse = 2(0.5) = 1.0 m
Example 2: Finding cable length from a known height
Imagine a ladder setup where the geometry matches a 30-60-90 triangle. If you know the long leg (the side opposite 60°) is 3 ft, the short leg and hypotenuse follow directly from the same ratio.
- Short leg = 3/√3 ft
- Hypotenuse = 2(3)/√3 ft
That gives you consistent measurements for materials and safer planning.
Common mistakes to avoid
- Mixing up the sides: The 30° side is the short leg, and the 60° side is the long leg.
- Using the wrong ratio: Don’t use 45-45-90 values (like √2) for a 30-60-90 triangle.
- Unit confusion: Always confirm the input unit and the output unit in the calculator.
Frequently Asked Questions
What are the side ratios in a 30 60 90 triangle?
In a 30-60-90 triangle, the sides are always in the ratio 1 : √3 : 2. The side opposite 30° is 1 (short leg), the side opposite 60° is √3 (long leg), and the side opposite 90° is 2 (hypotenuse).
If I know the hypotenuse, how do I find the short leg?
For a 30-60-90 triangle, the hypotenuse equals 2x where x is the short leg. So if your hypotenuse is H, the short leg is H/2. This comes directly from the fixed 1 : √3 : 2 ratio.
If I know the long leg, how do I find the hypotenuse?
The long leg equals x√3 while the hypotenuse equals 2x. If the long leg is L, then x = L/√3, so the hypotenuse is 2L/√3. This is exact and avoids rounding errors.
Why does the calculator include unit conversions?
Triangle ratios are unit-free, but real measurements use units like inches or meters. The calculator converts your input to the selected working unit, computes the side lengths, then converts results back. This prevents mistakes from manual unit conversions.
Is a 30 60 90 triangle always a right triangle?
Yes. By definition, a 30-60-90 triangle includes a 90° angle, so it is always right. The other two angles are fixed at 30° and 60°, which is why the 1 : √3 : 2 ratio always holds.
Quick checklist before you trust the results
- Confirm the triangle is truly 30-60-90 (angles 30°, 60°, 90°).
- Enter a positive number for the known side.
- Use the correct side type: short leg, long leg, or hypotenuse.
- Select the units you want for the outputs.
With those steps, the calculator gives the exact side lengths implied by the 30-60-90 ratio.
