What this calculator does
A 30-60-90 triangle is a right triangle whose three angles are fixed at 30°, 60° and 90°, which forces its three sides into a fixed ratio no matter how large or small the triangle is: 1 : √3 : 2, for the short leg, long leg and hypotenuse respectively. Knowing just one side length is enough to work out the other two.
This 30-60-90 triangle calculator takes whichever single side you know, the short leg, the long leg or the hypotenuse, and scales the fixed ratio to match it, then returns all three sides, the area and the perimeter.
The formula
The three sides always sit in the ratio short leg : long leg : hypotenuse = 1 : √3 : 2. Given any one side, the short leg is recovered first (dividing by the appropriate ratio factor), and the other two sides are then the short leg multiplied by √3 and by 2. Area follows as ½ × short leg × long leg, since the two legs meet at the right angle.
| Term | Meaning |
|---|---|
| Short leg | The side opposite the 30° angle, the shortest side. |
| Long leg | The side opposite the 60° angle, equal to the short leg × √3. |
| Hypotenuse | The side opposite the right angle, equal to the short leg × 2, and the longest side. |
The inputs explained
| Field | What to enter |
|---|---|
| Known side | Which side you are entering a length for. |
| Length of that known side | The length of that known side. |
When to use it
A geometry or trigonometry problem
Many textbook problems give one side of a 30-60-90 triangle and ask for the rest; this fills in the remaining sides directly from the fixed ratio rather than requiring trigonometric ratios to be looked up.
Roof pitches and ramps
A 30° or 60° incline is a common design angle, and knowing one measurement, such as the rise, lets the run and the slope length be worked out immediately from the fixed ratio.
Cutting a set-square or drafting template
A 30-60-90 set-square has this exact shape; scaling it to a specific size means computing all three sides from just one target length.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
All three sides as the short leg lengthens
The short leg (opposite the 30° angle) taken as the known side, across a range of lengths.
| Short leg | Long leg (opposite 60°) | Hypotenuse (opposite 90°) | Area |
|---|---|---|---|
| 1 | 1.732 | 2.000 | 0.8660 |
| 2 | 3.464 | 4.000 | 3.464 |
| 5 | 8.660 | 10.000 | 21.651 |
| 10 | 17.321 | 20.000 | 86.603 |
| 15 | 25.981 | 30.000 | 194.856 |
| 20 | 34.641 | 40.000 | 346.410 |
Solving from the hypotenuse instead
The hypotenuse taken as the known side, across a range of lengths.
| Hypotenuse | Short leg (opposite 30°) | Long leg (opposite 60°) | Area |
|---|---|---|---|
| 2 | 1.000 | 1.732 | 0.8660 |
| 4 | 2.000 | 3.464 | 3.464 |
| 10 | 5.000 | 8.660 | 21.651 |
| 20 | 10.000 | 17.321 | 86.603 |
| 30 | 15.000 | 25.981 | 194.856 |
| 40 | 20.000 | 34.641 | 346.410 |
Questions
What is the side ratio of a 30-60-90 triangle?
The three sides are always in the ratio 1 : √3 : 2, for the short leg (opposite 30°), the long leg (opposite 60°) and the hypotenuse (opposite 90°) respectively, regardless of the triangle's overall size.
How is a 30-60-90 triangle different from a 45-45-90 triangle?
A 30-60-90 triangle has three different angles and three different side lengths, in the ratio 1 : √3 : 2. A 45-45-90 triangle is isosceles, with two equal angles and two equal legs, in the ratio 1 : 1 : √2. They are two distinct special right triangles with different fixed ratios.
Where does the √3 in the ratio come from?
It comes from bisecting an equilateral triangle. Splitting an equilateral triangle with sides of length 2 down the middle gives two 30-60-90 triangles with a short leg of 1, a hypotenuse of 2, and, by Pythagoras, a long leg of √(2² − 1²) = √3.
Can I find the sides from just an area or perimeter instead of a side length?
Not directly with this calculator, which solves from one known side. Given only the area or perimeter, the short leg can be found algebraically first (since both area and perimeter are fixed functions of the short leg in this ratio), and that value can then be entered here to get every other measurement.
For the other common special right triangle, with two equal legs, see the isosceles right triangle calculator. For a right triangle with no special angle constraint, see the scalene triangle calculator.