30 60 90 Triangle Calculator
Solve a 30-60-90 right triangle from any single side. Enter the short leg, long leg, or hypotenuse in any unit and the fixed 1 : √3 : 2 side ratio fills in the other two, plus area and perimeter.
Example: with Which side do you know? Short leg (opposite the 30° angle) · Side length 5 → Hypotenuse (2x): 10 units.
- Short leg (x, opposite 30°)5 units
- Long leg (x√3, opposite 60°)8.66 units
- Area (√3/2)x²21.651 square units
Computed by the calculator below using its default values. Change any input to see your own numbers.
Sides are always in the ratio 1 : √3 : 2 — short leg : long leg : hypotenuse. It is half an equilateral triangle, which is why the ratio never changes.
Why the sides are always 1 : √3 : 2
A 30-60-90 triangle is exactly half an equilateral triangle. Slice an equilateral triangle with side 2x along its height and you get a right triangle whose hypotenuse is the original side (2x), whose short leg is half the base (x), and whose long leg is the height — by the Pythagorean theorem, √((2x)² − x²) = x√3. The angles 30°, 60°, and 90° lock those proportions in place, so one side determines everything.
The mapping trips people up in one specific way: the short leg sits opposite the small 30° angle, and the long leg sits opposite the 60° angle. So from a known long leg you divide by √3 (≈1.732) to get x, and from a hypotenuse you halve it. Drafting triangles, roof rafters at a 30° pitch, and trig-class ladder problems all reduce to these three multiplications.
How it’s calculated
Sides in ratio 1 : √3 : 2 with √3 ≈ 1.7320508. From short leg x: long = x√3, hypotenuse = 2x. From long leg L: x = L/√3. From hypotenuse h: x = h/2. Area = ½ · x · x√3 = (√3/2)x² ≈ 0.866025x²; perimeter = (3 + √3)x ≈ 4.732051x.
Angles must actually be 30-60-90 — for any other right triangle use the Pythagorean theorem or trig, because this fixed ratio no longer applies.
30-60-90 dimensions for common inputs
| Short leg (x) | Long leg (x√3) | Hypotenuse (2x) | Area |
|---|---|---|---|
| 1 | 1.732 | 2 | 0.87 |
| 2 | 3.464 | 4 | 3.46 |
| 3 | 5.196 | 6 | 7.79 |
| 4 | 6.928 | 8 | 13.86 |
| 5 | 8.660 | 10 | 21.65 |
| 6 | 10.392 | 12 | 31.18 |
| 10 | 17.321 | 20 | 86.60 |
Computed with the 1 : √3 : 2 ratio and A = (√3/2)x²; rounded.
Common mistakes
- Swapping the legs — the short leg faces the 30° angle, the long leg faces the 60° angle. Assigning them backwards skews every result by a factor of √3.
- Multiplying the hypotenuse by √3 to get a leg; the hypotenuse relates to the short leg by 2, and only the short leg relates to the long leg by √3.
- Using 1.7 for √3 in multi-step work — carry 1.7320508 and round once at the end.
- Applying the ratio to a right triangle that merely looks steep; without a true 30° angle the 1 : √3 : 2 shortcut is invalid.
Frequently asked questions
What is the 30 60 90 triangle ratio?
Short leg : long leg : hypotenuse = 1 : √3 : 2, or x, x√3, and 2x. Given any one side, divide by its ratio number to find x, then multiply out the other two.
Which side is opposite the 60 degree angle?
The long leg, x√3 ≈ 1.732x. Bigger angles face bigger sides: 30° faces x, 60° faces x√3, and the 90° right angle faces the hypotenuse 2x.
How do I solve it from the long leg?
Divide the long leg by √3 to get the short leg, then double the short leg for the hypotenuse. A 9-unit long leg gives x = 5.196 and a hypotenuse of 10.392.
Why is a 30-60-90 triangle half an equilateral triangle?
Cutting an equilateral triangle along its height bisects the 60° apex into 30°, leaves one full 60° corner, and creates a 90° foot — and the hypotenuse of each half is an untouched original side.
What are the area and perimeter formulas?
With short leg x: area = (√3/2)x² ≈ 0.866x² and perimeter = (3 + √3)x ≈ 4.732x. For x = 5 that is 21.65 square units and 23.66 units around.